Quadrilaterals and Polygons Alıştırma Soruları — GRE General Test

Soru 1Soru →

In convex quadrilateral $ABCD$ , diagonal $AC$ is drawn. It is given that $AB = 9$ , $BC = 12$ , $CD = 8$ , and $DA = 15$ , with $\angle ABC = 90^\circ$ . Which of the following statements must be true? Select all that apply.

Geçerli olan tümünü seçin

The length of diagonal

AC

is 15.

The area of quadrilateral

ABCD

is 114.

The perimeter of quadrilateral

ABCD

is 44.

A triangle can be formed using side lengths of 8, 15, and 32.

The sum of the interior angles of quadrilateral

ABCD

is

360^\circ

.

Cevabı ve açıklamayı göster

Cevap: The length of diagonal $AC$ is 15.; The perimeter of quadrilateral $ABCD$ is 44.; The sum of the interior angles of quadrilateral $ABCD$ is $360^\circ$ .

Cevap

The correct statements are that the length of diagonal

AC

is 15, the perimeter of quadrilateral

ABCD

is 44, and the sum of the interior angles of quadrilateral

ABCD

is

360^\circ

.

The statement specifying that the length of diagonal

AC

is 15 is correct because right triangle

ABC

has leg lengths 9 and 12, giving hypotenuse

\sqrt{81 + 144} = 15

. The statement specifying that the perimeter is 44 is correct because summing the outer side lengths yields

9 + 12 + 8 + 15 = 44

. The statement specifying that the interior angle sum is

360^\circ

is correct because every convex quadrilateral has an interior angle sum of

(4 - 2) \times 180^\circ = 360^\circ

.

Adım Adım Çözüm

1

Calculate the length of diagonal

AC

using triangle

ABC

.

AC = \sqrt{AB^2 + BC^2} = \sqrt{9^2 + 12^2} = 15

.

Since

\angle ABC = 90^\circ

, triangle

ABC

is a right triangle, allowing the application of the Pythagorean theorem.

2

Calculate the perimeter of quadrilateral

ABCD

.

Perimeter

= 9 + 12 + 8 + 15 = 44

.

The perimeter of a polygon is the sum of all its outer side lengths.

3

Verify the sum of the interior angles for quadrilateral

ABCD

.

Interior angle sum

= (4 - 2) \times 180^\circ = 360^\circ

.

The formula

(n - 2) \times 180^\circ

applies to all convex polygons.

4

Evaluate the incorrect claims regarding area and triangle formation.

Triangle

ADC

is not a right triangle (

8^2 + 15^2 \neq 15^2

), so its area is not 60; and side lengths 8, 15, and 32 violate the triangle inequality theorem (

8 + 15 = 23 < 32

).

Right triangle formulas require a right angle, and valid triangle side lengths must satisfy the triangle inequality theorem.

Anahtar Kavram

Properties of convex quadrilaterals, right triangle side relationships, and triangle inequality bounds

Tahmini Süre:1m 30s

Soru 2Soru →

In regular octagon $ABCDEFGH$ with side length $2$ , diagonals $AD$ and $BE$ intersect at point $P$ . What is the area of quadrilateral $ABCP$ ?

2 + \sqrt{2}

2\sqrt{2}

4

4 + \sqrt{2}

1 + 2\sqrt{2}

Cevabı ve açıklamayı göster

Cevap: $2 + \sqrt{2}$

Cevap

2 + \sqrt{2}

In regular octagon

ABCDEFGH

, each interior angle measures

135^\circ

. Segment

ABCD

forms an isosceles trapezoid where

AD

is parallel to

BC

. The perpendicular distance from

AD

to

BC

is

2 \sin(45^\circ) = \sqrt{2}

, and

AD = 2 + 2(2 \cos(45^\circ)) = 2 + 2\sqrt{2}

. Diagonal

BE

is parallel to side

CD

, so

\angle CBE = 45^\circ

, which implies

\angle ABE = 135^\circ - 45^\circ = 90^\circ

. Triangle

\triangle ABP

is therefore a right isosceles triangle with legs

AB = 2

and

BP = 2

, yielding hypotenuse

AP = 2\sqrt{2}

. Quadrilateral

ABCP

is a trapezoid with parallel sides

BC = 2

and

AP = 2\sqrt{2}

and height

h = \sqrt{2}

. Its area is

\frac{2 + 2\sqrt{2}}{2} \times \sqrt{2} = 2 + \sqrt{2}

.

Adım Adım Çözüm

1

Calculate the interior angle of a regular octagon

Interior angle =

\frac{(8-2) \times 180^\circ}{8} = 135^\circ

Regular polygon interior angle formula.

2

Determine the orientation and length of diagonal

AD

AD \parallel BC

, height between them is

h = 2 \sin(45^\circ) = \sqrt{2}

, and base

AD = 2 + 2(2 \cos(45^\circ)) = 2 + 2\sqrt{2}

Quadrilateral

ABCD

forms an isosceles trapezoid with side length

2

and interior angles

135^\circ

.

3

Determine the direction of diagonal

BE

and find intersection point

P

\angle CBE = 180^\circ - 135^\circ = 45^\circ

, making

\angle ABE = 135^\circ - 45^\circ = 90^\circ

Quadrilateral

BCDE

is an isosceles trapezoid with

BE \parallel CD

.

4

Calculate the dimensions of right triangle

\triangle ABP

and segment

AP

\triangle ABP

is a right triangle at

B

with

\angle BAP = 45^\circ

,

AB = 2

,

BP = 2

, and hypotenuse

AP = 2\sqrt{2}

Since

AP \parallel BC

, alternate interior angle relationships yield

\angle BAP = 45^\circ

.

5

Calculate the area of quadrilateral

ABCP

\text{Area}(ABCP) = \frac{BC + AP}{2} \times h = \frac{2 + 2\sqrt{2}}{2} \times \sqrt{2} = (1 + \sqrt{2})\sqrt{2} = 2 + \sqrt{2}

Quadrilateral

ABCP

is a trapezoid with parallel bases

BC = 2

and

AP = 2\sqrt{2}

and height

\sqrt{2}

.

Anahtar Kavram

Properties of regular polygons, isosceles trapezoids, diagonal angles, and area decomposition.

Tahmini Süre:2m 30s

Soru 3Soru →

In the $xy$ -plane, $A(1, 2)$ , $B(7, 4)$ , and $D(3, 8)$ are three vertices of rhombus $ABCD$ . A line $k$ passes through the origin $(0, 0)$ and bisects the area of rhombus $ABCD$ . What is the slope of line $k$ ?

\frac{6}{5}

\frac{5}{6}

1

-1

\frac{3}{4}

Cevabı ve açıklamayı göster

Cevap: $\frac{6}{5}$

Cevap

The slope of line

k

is

\frac{6}{5}

.

Any line that divides a parallelogram or rhombus into two equal areas must pass through its center of symmetry, which is the midpoint of its diagonals. The midpoint of diagonal

BD

with endpoints

(7, 4)

and

(3, 8)

is

\left(\frac{7+3}{2}, \frac{4+8}{2}\right) = (5, 6)

. Since line

k

passes through the origin

(0, 0)

and

(5, 6)

, its slope is

\frac{6 - 0}{5 - 0} = \frac{6}{5}

.

Adım Adım Çözüm

1

Identify the key geometric property of area-bisecting lines for parallelograms and rhombuses.

Any line that bisects the area of a rhombus must pass through its center of symmetry (the intersection point of its diagonals).

A rhombus is centrally symmetric about the intersection point of its diagonals, so any line through this point divides the rhombus into two congruent regions.

2

Find the coordinates of the center of symmetry by calculating the midpoint of diagonal

BD

.

Midpoint

M = \left(\frac{7+3}{2}, \frac{4+8}{2}\right) = (5, 6)

.

Opposite vertices

B(7,4)

and

D(3,8)

define one of the diagonals of rhombus

ABCD

.

3

Calculate the slope of line

k

passing through the origin

(0, 0)

and center point

M(5, 6)

.

\text{Slope } m = \frac{6 - 0}{5 - 0} = \frac{6}{5}.

The slope formula between

(x_1, y_1)

and

(x_2, y_2)

is

\frac{y_2 - y_1}{x_2 - x_1}

.

Anahtar Kavram

Center of Symmetry and Area Bisectors of Quadrilaterals

Tahmini Süre:2m 0s

Soru 4Soru →

In right trapezoid $ABCD$ , segment $AB$ is parallel to segment $CD$ , $\angle DAB = 90^\circ$ , $AD = 12$ , $CD = 15$ , and $BC = 13$ . Point $E$ lies on segment $CD$ such that quadrilateral $ABCE$ is a parallelogram. What is the perimeter of triangle $ADE$ ?

Cevabı ve açıklamayı göster

Cevap: 30

Cevap

30

Decomposing right trapezoid

ABCD

by dropping a perpendicular from

B

to

CD

forms a right triangle with height

12

and hypotenuse

13

. The Pythagorean theorem gives the base of this right triangle as

\sqrt{13^2 - 12^2} = 5

. Subtracting this from

CD = 15

yields

AB = 10

. Because

ABCE

is a parallelogram,

CE = AB = 10

, which leaves

DE = CD - CE = 15 - 10 = 5

. Triangle

ADE

is a right triangle with legs

AD = 12

and

DE = 5

, giving hypotenuse

AE = \sqrt{12^2 + 5^2} = 13

. The perimeter of triangle

ADE

is

12 + 5 + 13 = 30

.

Adım Adım Çözüm

1

Calculate the horizontal projection of segment

BC

onto base

CD

The length of the horizontal projection is

\sqrt{13^2 - 12^2} = 5

Segment

AD = 12

defines the perpendicular distance between parallel lines

AB

and

CD

2

Determine the length of parallel base

AB

AB = 15 - 5 = 10

The total length of base

CD = 15

is the sum of

AB

and the horizontal projection of slant side

BC

3

Calculate the length of segment

DE

DE = 15 - 10 = 5

Quadrilateral

ABCE

is a parallelogram, which implies

CE = AB = 10

4

Compute the hypotenuse

AE

and the total perimeter of triangle

ADE

AE = \sqrt{12^2 + 5^2} = 13

, so

\text{Perimeter} = 12 + 5 + 13 = 30

Triangle

ADE

is a right-angled triangle with right angle at vertex

D

Anahtar Kavram

Trapezoid height decomposition, parallelogram side properties, and Pythagorean theorem application

Soru 5Soru →

A regular polygon has interior angles that each measure $140^\circ$ . What is the total number of diagonals that can be drawn in this polygon?

9

14

27

36

54

Cevabı ve açıklamayı göster

Cevap: 27

Cevap

The total number of diagonals that can be drawn in the polygon is 27.

To find the total number of diagonals in a regular polygon, first determine its number of sides

n

. Using the interior angle formula

\frac{(n-2) \times 180^\circ}{n} = 140^\circ

, we find

n = 9

. Next, applying the diagonal formula

D = \frac{n(n-3)}{2}

for

n = 9

gives

D = \frac{9 \times 6}{2} = 27

. Thus, the value 27 is correct.

Adım Adım Çözüm

1

Find the number of sides

n

of the regular polygon using the interior angle measure.

The polygon has

n = 9

sides.

Each interior angle of a regular

n

-gon is given by

\frac{(n - 2) \times 180^\circ}{n} = 140^\circ

. Solving for

n

:

180n - 360 = 140n \implies 40n = 360 \implies n = 9

.

2

Calculate the total number of diagonals using the formula

D = \frac{n(n - 3)}{2}

.

The total number of diagonals is 27.

Substituting

n = 9

into the formula yields

D = \frac{9(9 - 3)}{2} = \frac{9 \times 6}{2} = 27

.

Anahtar Kavram

Interior Angle Measure and Diagonal Formula for Regular Polygons

Tahmini Süre:1m 0s

Soru 6Soru →

A rectangle has a length of $8$ units and a width of $6$ units. Which of the following statements about this rectangle must be true? Select all that apply.

Geçerli olan tümünü seçin

The area of the rectangle is

48

square units.

The length of each diagonal of the rectangle is

10

units.

The perimeter of the rectangle is

28

units.

The length of each diagonal of the rectangle is

14

units.

The area of the rectangle is

28

square units.

Cevabı ve açıklamayı göster

Cevap: The area of the rectangle is $48$ square units.; The length of each diagonal of the rectangle is $10$ units.; The perimeter of the rectangle is $28$ units.

Cevap

The correct statements are that the area of the rectangle is 48 square units, the length of each diagonal is 10 units, and the perimeter of the rectangle is 28 units.

The statement regarding the area being 48 square units is correct because

8 \times 6 = 48

. The statement regarding the diagonal being 10 units is correct because

\sqrt{8^2 + 6^2} = 10

. The statement regarding the perimeter being 28 units is correct because

2 \times (8 + 6) = 28

.

Adım Adım Çözüm

1

Calculate the area of the rectangle

\text{Area} = 8 \times 6 = 48

square units

The area formula for a rectangle is length multiplied by width.

2

Calculate the length of the diagonal using the Pythagorean theorem

\text{Diagonal} = \sqrt{8^2 + 6^2} = \sqrt{64 + 36} = 10

units

The sides and diagonal of a rectangle form a right triangle where the diagonal is the hypotenuse.

3

Calculate the perimeter of the rectangle

\text{Perimeter} = 2 \times (8 + 6) = 28

units

The perimeter formula for a rectangle is twice the sum of its length and width.

Anahtar Kavram

Basic geometric properties of rectangles including area, perimeter, and diagonal calculation via the Pythagorean theorem.

Soru 7Soru →

Trapezoid $PQRS$ has parallel sides $PQ$ and $RS$ with lengths of $7$ centimeters and $13$ centimeters, respectively. If the perpendicular height between these parallel sides is $4$ centimeters, what is the area of trapezoid $PQRS$ , in square centimeters?

24

28

40

52

80

Cevabı ve açıklamayı göster

Cevap: $40$

Cevap

The area of trapezoid

PQRS

is

40

square centimeters.

The area of a trapezoid is found by averaging the lengths of the two parallel bases and multiplying by the perpendicular height:

\text{Area} = \frac{7 + 13}{2} \times 4 = 10 \times 4 = 40

square centimeters.

Adım Adım Çözüm

1

Identify the formula for the area of a trapezoid

\text{Area} = \frac{b_1 + b_2}{2} \times h

, where

b_1

and

b_2

are the lengths of the parallel bases and

h

is the height.

The area of any trapezoid is equal to the average of its parallel bases multiplied by its perpendicular height.

2

Substitute the given dimensions into the formula

\text{Area} = \frac{7 + 13}{2} \times 4

The given bases are

b_1 = 7\text{ cm}

and

b_2 = 13\text{ cm}

, and the height is

h = 4\text{ cm}

.

3

Calculate the average base length and multiply by the height

\frac{20}{2} \times 4 = 10 \times 4 = 40\text{ cm}^2

Simplifying

\frac{7 + 13}{2}

gives

10

, and

10 \times 4 = 40

.

Anahtar Kavram

Trapezoid Area Formula

Tahmini Süre:45s

Soru 8Soru →

The sum of the measures of all interior angles of a convex polygon, excluding one interior angle $\theta$ , is equal to $2190^\circ$ . If the degree measure of $\theta$ is an integer, what is the perimeter of a regular polygon with $n$ sides, where $n$ is the number of sides of the original polygon and each side length is $\frac{\theta}{10}$ units?

195

210

225

240

255

Cevabı ve açıklamayı göster

Cevap: 225

Cevap

The perimeter of the regular polygon is 225.

The sum of the interior angles of a convex polygon with

n

sides is

(n-2) \times 180^\circ

. Adding the excluded interior angle

\theta

to

2190^\circ

yields the total sum

S

. Because

0^\circ < \theta < 180^\circ

,

S

must fall strictly between

2190^\circ

and

2370^\circ

. The only multiple of

180^\circ

in this interval is

2340^\circ

. Setting

(n-2)\times 180^\circ = 2340^\circ

yields

n = 15

. Solving for

\theta

gives

\theta = 2340^\circ - 2190^\circ = 150^\circ

. The side length is

\frac{150}{10} = 15

, making the perimeter

15 \times 15 = 225

.

Adım Adım Çözüm

1

Set up the inequality for the sum of interior angles of a convex polygon.

The total sum of interior angles for a convex

n

-gon is

S = (n-2) \times 180^\circ

. Given

S - \theta = 2190^\circ

, we have

S = 2190^\circ + \theta

.

The sum of interior angles of any convex polygon with

n

sides is

(n-2) \times 180^\circ

.

2

Determine the value of

S

using the bounds for an interior angle of a convex polygon.

Since

0^\circ < \theta < 180^\circ

, it follows that

2190^\circ < S < 2190^\circ + 180^\circ = 2370^\circ

. The only multiple of

180^\circ

in this range is

2340^\circ

.

S

must be an integer multiple of

180^\circ

and

\theta

must be strictly between

0^\circ

and

180^\circ

.

3

Calculate the number of sides

n

and the missing angle

\theta

.

(n-2) \times 180^\circ = 2340^\circ \implies n - 2 = 13 \implies n = 15

. Then

\theta = 2340^\circ - 2190^\circ = 150^\circ

.

Solving the linear equations gives exact values for the number of sides and the excluded angle.

4

Compute the perimeter of the regular regular

n

-gon.

Side length

= \frac{\theta}{10} = \frac{150}{10} = 15

. Perimeter

= n \times \text{side length} = 15 \times 15 = 225

.

The perimeter of a regular polygon is the product of its number of sides and its individual side length.

Anahtar Kavram

Sum of Interior Angles of Convex Polygons

Tahmini Süre:2m 0s

Soru 9Soru →

In convex quadrilateral $ABCD$ , diagonals $AC$ and $BD$ intersect at point $E$ . The area of $\triangle ABE$ is $4$ , the area of $\triangle BCE$ is $8$ , and the area of $\triangle CDE$ is $16$ . Which of the following statements must be true? Select all that apply.

Geçerli olan tümünü seçin

The area of

\triangle DAE

is

8

.

Quadrilateral

ABCD

is a trapezoid with

AB

parallel to

CD

.

The length of segment

CD

is twice the length of segment

AB

.

The total area of quadrilateral

ABCD

is

32

.

The midpoint of diagonal

AC

lies on segment

DE

.

Cevabı ve açıklamayı göster

Cevap: The area of $\triangle DAE$ is $8$ .; Quadrilateral $ABCD$ is a trapezoid with $AB$ parallel to $CD$ .; The length of segment $CD$ is twice the length of segment $AB$ .

Cevap

The correct statements are that the area of triangle DAE is 8, quadrilateral ABCD is a trapezoid with AB parallel to CD, and the length of segment CD is twice the length of segment AB.

The statements confirming that the area of triangle DAE is 8, that ABCD is a trapezoid with AB parallel to CD, and that CD is twice length AB are all derived using area ratio properties of intersecting diagonals and triangle similarity.

Adım Adım Çözüm

1

Determine the area of triangle DAE using diagonal segment ratios.

Area of triangle DAE = 8.

Triangles sharing a base line have areas proportional to the segments into which the intersecting line divides that base line: AE/EC = Area(ABE)/Area(BCE) = 4/8 = 1/2, so Area(DAE) = (1/2) * 16 = 8.

2

Check parallelism of opposite sides AB and CD.

AB is parallel to CD, making ABCD a trapezoid.

Area(ABC) = 4 + 8 = 12 and Area(ABD) = 4 + 8 = 12. Triangles with equal areas on the common base AB must have equal heights, implying line CD is parallel to line AB.

3

Calculate side ratio CD / AB using similar triangles.

CD = 2 * AB.

Since AB || CD, triangle ABE is similar to triangle CDE. The ratio of their areas is 16/4 = 4, so the side length ratio CD/AB = sqrt(4) = 2.

4

Compute total area of quadrilateral ABCD and evaluate diagonal midpoint position.

Total area is 36 (not 32), and the midpoint of AC is at (1/2)AC from A, which is distinct from E at (1/3)AC.

Total area = 4 + 8 + 16 + 8 = 36. Since E divides AC in a 1:2 ratio, E is not the midpoint of AC.

Anahtar Kavram

Properties of convex quadrilaterals, area decomposition via diagonal ratios, trapezoid parallelism criteria, and triangle similarity.

Soru 10Soru →

The ratio of the measure of each interior angle of a regular $n$ -sided polygon $P$ to the measure of each interior angle of a regular $(n+2)$ -sided polygon is $\frac{24}{25}$ . Which of the following statements about polygon $P$ must be true? Select all such statements.

Geçerli olan tümünü seçin

Polygon

P

has

10

sides.

Polygon

P

has

35

diagonals.

The measure of each exterior angle of polygon

P

is

40^\circ

.

The sum of the measures of the interior angles of polygon

P

is

1{}800^\circ

.

The measure of each interior angle of polygon

P

is

144^\circ

.

Cevabı ve açıklamayı göster

Cevap: Polygon $P$ has $10$ sides.; Polygon $P$ has $35$ diagonals.; The measure of each interior angle of polygon $P$ is $144^\circ$ .

Cevap

The statements asserting that polygon P has 10 sides, polygon P has 35 diagonals, and each interior angle of polygon P measures 144 degrees are all correct.

To determine which statements are true, we set up the ratio of the interior angle of a regular

n

-sided polygon to that of a regular

(n+2)

-sided polygon:

\frac{(n-2)(n+2)}{n^2} = \frac{24}{25}

. Simplifying gives

1 - \frac{4}{n^2} = \frac{24}{25}

, which leads to

n^2 = 100

, so

n = 10

. Therefore, polygon

P

is a regular decagon (10 sides). Evaluating the properties of a regular 10-gon shows that the number of diagonals is

\frac{10(10-3)}{2} = 35

, each interior angle measures

\frac{8 \cdot 180^\circ}{10} = 144^\circ

, each exterior angle measures

\frac{360^\circ}{10} = 36^\circ

, and the sum of the interior angles is

8 \cdot 180^\circ = 1{}440^\circ

. Consequently, the options stating that the polygon has 10 sides, has 35 diagonals, and has interior angles measuring

144^\circ

are correct.

Adım Adım Çözüm

1

Set up the algebraic equation comparing the interior angles of an

n

-gon and an

(n+2)

-gon.

\frac{\frac{(n-2) \cdot 180^\circ}{n}}{\frac{n \cdot 180^\circ}{n+2}} = \frac{24}{25} \implies \frac{(n-2)(n+2)}{n^2} = \frac{24}{25}

The formula for each interior angle of a regular polygon with

k

sides is

\frac{(k-2) \cdot 180^\circ}{k}

.

2

Solve for

n

.

1 - \frac{4}{n^2} = \frac{24}{25} \implies \frac{4}{n^2} = \frac{1}{25} \implies n^2 = 100 \implies n = 10

Expanding

(n-2)(n+2) = n^2 - 4

allows simplifying the algebraic ratio.

3

Evaluate polygon properties for

n = 10

.

Diagonals:

\frac{10(10-3)}{2} = 35

; Interior angle:

\frac{8 \cdot 180^\circ}{10} = 144^\circ

; Exterior angle:

\frac{360^\circ}{10} = 36^\circ

; Interior angle sum:

8 \cdot 180^\circ = 1{}440^\circ

.

Apply standard formulas for diagonal count, exterior angle measure, and interior angle sum for a regular decagon.

Anahtar Kavram

Interior and exterior angle formulas of regular polygons and diagonal counting formulas

Soru 11Soru →

In convex quadrilateral $ABCD$ , the diagonals $AC$ and $BD$ intersect at point $P$ at right angles ( $AC \perp BD$ ). The length of diagonal $AC$ is $16$ and the length of diagonal $BD$ is $12$ . Points $E$ , $F$ , $G$ , and $H$ are the midpoints of sides $AB$ , $BC$ , $CD$ , and $DA$ , respectively. Which of the following statements MUST be true? Select all such statements.

Geçerli olan tümünü seçin

The perimeter of quadrilateral

EFGH

is

28

.

Quadrilateral

EFGH

is a rectangle.

The area of quadrilateral

ABCD

is

96

.

The area of quadrilateral

EFGH

is

96

.

Quadrilateral

EFGH

is a rhombus.

Cevabı ve açıklamayı göster

Cevap: The perimeter of quadrilateral $EFGH$ is $28$ .; Quadrilateral $EFGH$ is a rectangle.; The area of quadrilateral $ABCD$ is $96$ .

Cevap

The true statements are that the perimeter of quadrilateral

EFGH

is

28

, quadrilateral

EFGH

is a rectangle, and the area of quadrilateral

ABCD

is

96

.

Applying the Midpoint Theorem shows that midsegments

EF

and

GH

are parallel to

AC

with length

8

, while

FG

and

HE

are parallel to

BD

with length

6

. This gives a perimeter of

8 + 6 + 8 + 6 = 28

. Because

AC \perp BD

, the adjacent midsegments meet at

90^\circ

, confirming that quadrilateral

EFGH

is a rectangle. Additionally, for any orthodiagonal quadrilateral, the area is

\frac{1}{2} d_1 d_2 = \frac{1}{2} \times 16 \times 12 = 96

.

Adım Adım Çözüm

1

Determine the side lengths of midpoint quadrilateral

EFGH

using the Triangle Midpoint Theorem.

EF = GH = \frac{1}{2}AC = 8

and

FG = HE = \frac{1}{2}BD = 6

.

In any triangle, the segment connecting the midpoints of two sides is parallel to the third side and half its length.

2

Calculate the perimeter of quadrilateral

EFGH

.

Perimeter =

EF + FG + GH + HE = 8 + 6 + 8 + 6 = 28

.

The perimeter is the sum of all four side lengths of the quadrilateral.

3

Determine the shape classification of quadrilateral

EFGH

.

EFGH

is a rectangle.

Since

EF \parallel AC

and

FG \parallel BD

, the angle between

EF

and

FG

equals the angle between diagonals

AC

and

BD

. Given

AC \perp BD

, the angle

\angle EFG = 90^\circ

. Opposite sides are equal and parallel with right angles, making

EFGH

a rectangle (and not a rhombus, as adjacent sides

8 \neq 6

).

4

Calculate the area of quadrilateral

ABCD

and quadrilateral

EFGH

.

Area(

ABCD

) =

\frac{1}{2} \times AC \times BD = \frac{1}{2} \times 16 \times 12 = 96

; Area(

EFGH

) =

8 \times 6 = 48

.

The area of a quadrilateral with perpendicular diagonals is half the product of its diagonal lengths. The midpoint quadrilateral has half the area of the outer quadrilateral.

Anahtar Kavram

Midpoint Theorem (Varignon's Theorem) and Area of Orthodiagonal Quadrilaterals

Tahmini Süre:2m 0s

Soru 12Soru →

In convex quadrilateral $ABCD$ , $\angle ABC = 90^\circ$ and $\angle ADC = 90^\circ$ . If $AB = BC$ , $AD = 6$ , and $CD = 8$ , what is the area of quadrilateral $ABCD$ ?

48

49

55

73

74

Cevabı ve açıklamayı göster

Cevap: 49

Cevap

49

The correct answer is 49. Dividing quadrilateral

ABCD

along diagonal

AC

creates two right triangles:

\triangle ADC

with legs 6 and 8, and

\triangle ABC

with hypotenuse

AC

and equal legs. Using the Pythagorean theorem on

\triangle ADC

gives hypotenuse

AC = \sqrt{6^2 + 8^2} = 10

, and its area is

\frac{1}{2}(6)(8) = 24

. For isosceles right triangle

\triangle ABC

,

AB^2 + BC^2 = 10^2 \implies 2(AB^2) = 100 \implies AB^2 = 50

, so its area is

\frac{1}{2}(50) = 25

. Adding both triangle areas yields

24 + 25 = 49

.

Adım Adım Çözüm

1

Divide the quadrilateral into two right triangles using diagonal

AC

.

Quadrilateral

ABCD

is split into

\triangle ADC

and

\triangle ABC

, both of which are right-angled triangles sharing hypotenuse

AC

.

Diagonal

AC

connects the vertices opposite the

90^\circ

angles.

2

Calculate the length of diagonal

AC

using

\triangle ADC

.

AC^2 = AD^2 + CD^2 = 6^2 + 8^2 = 36 + 64 = 100 \implies AC = 10

.

\triangle ADC

is a right triangle with legs of length 6 and 8.

3

Find the area of

\triangle ADC

.

\text{Area}(\triangle ADC) = \frac{1}{2} \times AD \times CD = \frac{1}{2} \times 6 \times 8 = 24

.

The area of a right triangle is half the product of its legs.

4

Determine the area of isosceles right triangle

\triangle ABC

.

Let

AB = BC = s

. Since

s^2 + s^2 = AC^2

, we have

2s^2 = 100 \implies s^2 = 50

. Thus,

\text{Area}(\triangle ABC) = \frac{1}{2} s^2 = \frac{1}{2} \times 50 = 25

.

\triangle ABC

is a right triangle with equal legs

s

and hypotenuse

AC = 10

.

5

Sum the areas of the two triangles to get the total area.

\text{Total Area} = 24 + 25 = 49

.

The total area of the quadrilateral is the sum of the areas of its non-overlapping constituent triangles.

Anahtar Kavram

Polygon area decomposition using diagonal partitioning and the Pythagorean theorem

Tahmini Süre:1m 30s

Soru 13Soru →

In convex pentagon $ABCDE$ , the measure of interior angle $\angle A$ is $100^\circ$ . The measures of the remaining four interior angles, $\angle B$ , $\angle C$ , $\angle D$ , and $\angle E$ , are in the ratio $2 : 3 : 3 : 3$ . What is the measure, in degrees, of $\angle B$ ?

40^\circ

80^\circ

108^\circ

120^\circ

160^\circ

Cevabı ve açıklamayı göster

Cevap: $80^\circ$

Cevap

The correct answer is

80^\circ

, which corresponds to the measure of

\angle B

.

The sum of interior angles of a 5-sided polygon (pentagon) is

(5-2) \times 180^\circ = 540^\circ

. Subtracting

\angle A = 100^\circ

leaves

440^\circ

for the remaining four angles. The total ratio parts for these four angles is

2 + 3 + 3 + 3 = 11

. Dividing

440^\circ

by 11 gives

40^\circ

per ratio unit. Since

\angle B

corresponds to 2 ratio parts, its measure is

2 \times 40^\circ = 80^\circ

.

Adım Adım Çözüm

1

Calculate the sum of all interior angles of the pentagon.

The sum of interior angles for an

n

-sided polygon is given by

(n - 2) \times 180^\circ

. For a pentagon (

n = 5

), the sum is

(5 - 2) \times 180^\circ = 3 \times 180^\circ = 540^\circ

.

Determining the total interior angle sum is necessary to find the sum of the unknown angles.

2

Subtract the known angle measure

\angle A

from the total interior angle sum.

The combined sum of angles

\angle B + \angle C + \angle D + \angle E = 540^\circ - 100^\circ = 440^\circ

.

Isolating the sum of the remaining four angles allows distribution according to the given ratio.

3

Determine the value of one ratio unit.

The sum of ratio parts is

2 + 3 + 3 + 3 = 11

parts. One part is equal to

440^\circ / 11 = 40^\circ

.

Finding the magnitude of a single ratio unit enables calculation of any individual angle.

4

Multiply the single unit value by the ratio coefficient for

\angle B

.

Since

\angle B

corresponds to 2 parts,

\angle B = 2 \times 40^\circ = 80^\circ

.

This yields the requested measure of angle

\angle B

.

Anahtar Kavram

Sum of interior angles of an n-sided polygon:

(n - 2) \times 180^\circ

, combined with proportional partitioning of angle sums.

Alternatif Yöntem

Express the angles in terms of a variable

x

. Let the remaining angles be

2x, 3x, 3x, 3x

. Set up the equation

100^\circ + 2x + 3x + 3x + 3x = 540^\circ

, simplify to

100 + 11x = 540

, solve for

x = 40^\circ

, and then evaluate

\angle B = 2x = 80^\circ

.

Tahmini Süre:1m 15s

Soru 14Soru →

In trapezoid $ABCD$ , side $AB$ is parallel to side $CD$ , and $\angle DAB = 90^\circ$ . If $AB = 14$ , $CD = 8$ , and $BC = 10$ , what is the area of trapezoid $ABCD$ ?

40

66

88

110

176

Cevabı ve açıklamayı göster

Cevap: 88

Cevap

88

By drawing an altitude from point

C

perpendicular to base

AB

at point

H

, the trapezoid is partitioned into a rectangle

ADCH

and a right triangle

CHB

. The base of triangle

CHB

is

HB = 14 - 8 = 6

, and its hypotenuse is

BC = 10

. Applying the Pythagorean theorem yields

CH = \sqrt{10^2 - 6^2} = 8

. Using the trapezoid area formula

\frac{b_1 + b_2}{2} \times h

gives

\frac{14 + 8}{2} \times 8 = 88

.

Adım Adım Çözüm

1

Decompose the trapezoid into a rectangle and a right triangle.

Draw a line segment from vertex

C

perpendicular to base

AB

, intersecting

AB

at point

H

. Quadrilateral

ADCH

is a rectangle, and triangle

CHB

is a right triangle with right angle at

H

.

Splitting the trapezoid allows the perpendicular height to be determined using right triangle properties.

2

Determine the horizontal base of right triangle

CHB

.

Since

ADCH

is a rectangle,

AH = CD = 8

. Therefore,

HB = AB - AH = 14 - 8 = 6

.

Opposite sides of a rectangle are equal in length.

3

Calculate height

CH

using the Pythagorean theorem.

In right triangle

CHB

,

CH^2 + HB^2 = BC^2 \implies CH^2 + 6^2 = 10^2 \implies CH^2 + 36 = 100 \implies CH = \sqrt{64} = 8

.

The height of the trapezoid corresponds to leg

CH

of right triangle

CHB

.

4

Calculate the area of trapezoid

ABCD

.

\text{Area} = \frac{AB + CD}{2} \times CH = \frac{14 + 8}{2} \times 8 = 11 \times 8 = 88.

The area formula for a trapezoid is half the sum of the parallel bases multiplied by the perpendicular height.

Anahtar Kavram

Decomposing a trapezoid into a rectangle and a right triangle to calculate height via the Pythagorean theorem

Tahmini Süre:1m 15s

Soru 15Soru →

An isosceles trapezoid has parallel base lengths of $10$ and $26$ , and an altitude of $15$ . A line segment connects the midpoints of the two non-parallel legs, dividing the figure into two smaller trapezoids. What is the area of the larger of these two smaller trapezoids?

Cevabı ve açıklamayı göster

Cevap: 165

Cevap

165

The midsegment of a trapezoid connects the midpoints of the non-parallel legs, and its length is the average of the two parallel bases:

\frac{10 + 26}{2} = 18

. Because the line connects midpoints, it also bisects the altitude, making the height of each smaller trapezoid equal to

\frac{15}{2} = 7.5

. The larger of the two resulting trapezoids has bases of lengths

18

and

26

. Using the trapezoid area formula

\text{Area} = \frac{b_1 + b_2}{2} \times h

, we obtain

\frac{18 + 26}{2} \times 7.5 = 22 \times 7.5 = 165

.

Adım Adım Çözüm

1

Calculate the length of the midsegment connecting the midpoints of the non-parallel legs.

The midsegment length is

18

.

The midsegment of a trapezoid is parallel to the bases and its length equals the average of the two base lengths:

\frac{10 + 26}{2} = 18

.

2

Determine the height of the smaller subtrapezoid.

The height of the subtrapezoid is

7.5

.

The segment connecting the midpoints of the legs bisects the overall altitude of

15

, giving a height of

\frac{15}{2} = 7.5

for each subtrapezoid.

3

Compute the area of the larger subtrapezoid.

The area is

165

.

The larger subtrapezoid is bounded by the midsegment (length

18

) and the bottom base (length

26

). Applying the trapezoid area formula yields

\text{Area} = \frac{18 + 26}{2} \times 7.5 = 22 \times 7.5 = 165

.

Anahtar Kavram

Trapezoid Midsegment Theorem and Subdivided Area Calculation

Soru 16Soru →

In a convex polygon, the sum of the measures of all interior angles except one is $1,200^\circ$ . If the measure of the remaining interior angle is an integer degree, what is the number of sides of the polygon?

7

8

9

10

11

Cevabı ve açıklamayı göster

Cevap: 9

Cevap

The number of sides of the polygon is 9.

The interior angle sum of an

n

-sided polygon is

(n - 2) \times 180^\circ

. Because the polygon is convex, the missing angle

x

must be between

0^\circ

and

180^\circ

. Adding

x

to

1,200^\circ

gives a total sum between

1,200^\circ

and

1,380^\circ

. The only multiple of

180^\circ

in this range is

1,260^\circ

, which corresponds to

(n - 2) = 7

, or

n = 9

sides.

Adım Adım Çözüm

1

Set up the inequality for the interior angle sum of a convex polygon

The sum of the interior angles of an

n

-sided convex polygon is

S = (n - 2) \times 180^\circ

. Let

x

be the measure of the remaining interior angle, where

0^\circ < x < 180^\circ

.

Every interior angle of a convex polygon must be strictly greater than

0^\circ

and strictly less than

180^\circ

.

2

Formulate the bounds for

(n - 2) \times 180^\circ

Since

S = 1,200^\circ + x

, we have

1,200^\circ < (n - 2) \times 180^\circ < 1,200^\circ + 180^\circ = 1,380^\circ

.

Adding the bounds of

x

(

0^\circ

to

180^\circ

) to the given sum of

1,200^\circ

establishes the valid range for the total angle sum.

3

Solve for the integer value of

n

Dividing the inequality by

180^\circ

gives

6.67 < n - 2 < 7.67

. Since

n - 2

must be an integer,

n - 2 = 7

, which means

n = 9

.

The only integer in the range

(6.67, 7.67)

is 7, corresponding to a 9-sided polygon with a remaining angle of

x = 1,260^\circ - 1,200^\circ = 60^\circ

.

Anahtar Kavram

Polygon Interior Angle Sum Theorem

Alternatif Yöntem

Divide

1,200^\circ

by

180^\circ

to get

6.67

. Since

(n - 2)

must be an integer and the remaining angle is positive and less than

180^\circ

, round

6.67

up to the next integer, 7. Thus,

n - 2 = 7

, so

n = 9

.

Tahmini Süre:1m 15s

Soru 17Soru →

In the $xy$ -plane, quadrilateral $ABCD$ has vertices $A(0, 0)$ , $B(6, 0)$ , $C(8, 4)$ , and $D(2, 4)$ . Which of the following statements must be true? Select all such statements.

Geçerli olan tümünü seçin

Quadrilateral

ABCD

is a parallelogram.

The area of quadrilateral

ABCD

is

24

.

The diagonals

AC

and

BD

bisect each other at the point

(4, 2)

.

The length of diagonal

AC

is equal to the length of diagonal

BD

.

The diagonals

AC

and

BD

are perpendicular to each other.

Cevabı ve açıklamayı göster

Cevap: Quadrilateral $ABCD$ is a parallelogram.; The area of quadrilateral $ABCD$ is $24$ .; The diagonals $AC$ and $BD$ bisect each other at the point $(4, 2)$ .

Cevap

The correct statements are that quadrilateral

ABCD

is a parallelogram, its area is

24

, and its diagonals bisect each other at

(4, 2)

.

Quadrilateral

ABCD

is a parallelogram because both pairs of opposite sides are congruent and parallel (

AB = DC = 6

along the horizontal line, and

AD = BC = \sqrt{20}

). The area is base times height, which is

6 \times 4 = 24

. The diagonals bisect each other at their common midpoint

(4, 2)

.

Adım Adım Çözüm

1

Determine side lengths and slopes to classify the quadrilateral

Side

AB

is horizontal with length

6

; side

DC

is horizontal with length

6

. Side

AD

has components

(2,4)

and length

\sqrt{20}

; side

BC

has components

(2,4)

and length

\sqrt{20}

. Since opposite sides are parallel and congruent,

ABCD

is a parallelogram.

To verify if the quadrilateral is a parallelogram.

2

Calculate the area of the quadrilateral

Area =

\text{base} \times \text{height} = 6 \times 4 = 24

.

To evaluate the area statement.

3

Find the midpoints and lengths of diagonals

AC

and

BD

Midpoint of

AC = (4, 2)

and midpoint of

BD = (4, 2)

, so they bisect each other. Length

AC = \sqrt{8^2 + 4^2} = \sqrt{80} = 4\sqrt{5}

, and length

BD = \sqrt{(-4)^2 + 4^2} = \sqrt{32} = 4\sqrt{2}

.

To check diagonal bisection and length equality.

4

Determine the slopes of the diagonals to check for perpendicularity

Slope of

AC = \frac{4}{8} = \frac{1}{2}

; slope of

BD = \frac{4}{-4} = -1

. Product of slopes

= \frac{1}{2} \times (-1) = -\frac{1}{2} \neq -1

, so they are not perpendicular.

To verify whether the diagonals intersect at right angles.

Anahtar Kavram

Properties of quadrilaterals in the coordinate plane, including parallelogram identification, area calculation, midpoint theorem for diagonals, and perpendicular slope test.

Tahmini Süre:1m 30s

Soru 18Soru →

In rhombus $ABCD$ , the perimeter is $52$ and the length of diagonal $AC$ is $10$ . What is the area of rhombus $ABCD$ ?

60

96

120

169

240

Cevabı ve açıklamayı göster

Cevap: 120

Cevap

The area of rhombus

ABCD

is 120.

Because all four sides of a rhombus are equal in length, a perimeter of 52 implies each side measures 13. The diagonals of a rhombus intersect at right angles and bisect each other. Given diagonal AC has a length of 10, half of AC is 5. Using the Pythagorean theorem on one of the right triangles formed by the intersecting diagonals gives a half-diagonal length of sqrt(13^2 - 5^2) = 12 for BD. Thus, the total length of diagonal BD is 24. The area of the rhombus is (1/2) * 10 * 24 = 120.

Adım Adım Çözüm

1

Find the side length of rhombus

ABCD

.

Side length

s = 52 / 4 = 13

.

A rhombus has four equal sides, so its perimeter divided by 4 gives the length of one side.

2

Use the properties of rhombus diagonals to find half of diagonal

BD

.

Half of

AC = 5

. Half of

BD = \sqrt{13^2 - 5^2} = \sqrt{169 - 25} = \sqrt{144} = 12

.

The diagonals of a rhombus bisect each other at right angles, forming four congruent right triangles whose hypotenuse is the side length (

13

) and legs are the half-diagonals.

3

Calculate the full length of diagonal

BD

.

Length of diagonal

BD = 2 \times 12 = 24

.

The full diagonal length is twice the length of its half.

4

Calculate the area of rhombus

ABCD

.

Area

= \frac{1}{2} \times d_1 \times d_2 = \frac{1}{2} \times 10 \times 24 = 120

.

The area of any rhombus is equal to half the product of its diagonal lengths.

Anahtar Kavram

Properties of a Rhombus: Perpendicular Bisecting Diagonals and Area Formula

Tahmini Süre:1m 30s

Soru 19Soru →

In a regular polygon, the ratio of the measure of an interior angle to the measure of an exterior angle is $7:2$ . What is the total number of diagonals of this polygon?

Cevabı ve açıklamayı göster

Cevap: 27

Cevap

The total number of diagonals of the regular polygon is

27

.

The correct answer is

27

. An interior angle and an exterior angle of a polygon are supplementary (

180^\circ

). Given the ratio

7:2

, the exterior angle is

\frac{2}{9} \times 180^\circ = 40^\circ

. Since the sum of exterior angles of any convex polygon is

360^\circ

, the number of sides is

n = \frac{360^\circ}{40^\circ} = 9

. Using the formula for the number of diagonals,

\frac{n(n-3)}{2}

, we obtain

\frac{9(9-3)}{2} = 27

.

Adım Adım Çözüm

1

Determine the measure of the exterior angle using the given interior-to-exterior ratio.

Exterior angle measure =

40^\circ

At any vertex of a polygon, the interior angle and exterior angle sum to

180^\circ

. With a ratio of

7:2

, the exterior angle represents

\frac{2}{7+2} = \frac{2}{9}

of the total

180^\circ

.

2

Calculate the number of sides (

n

) of the regular polygon.

n = 9

The sum of the exterior angles of any convex polygon is

360^\circ

. Since all exterior angles in a regular polygon are equal,

n = \frac{360^\circ}{40^\circ} = 9

.

3

Calculate the total number of diagonals using the formula

\frac{n(n-3)}{2}

.

Number of diagonals =

27

Substituting

n = 9

into

\frac{n(n-3)}{2}

yields

\frac{9 \times 6}{2} = 27

.

Anahtar Kavram

Interior and exterior angle properties of regular polygons, and the diagonal counting formula for convex polygons.

Soru 20Soru →

A convex hexagon has five interior angles measuring $115^\circ$ , $125^\circ$ , $130^\circ$ , $140^\circ$ , and $150^\circ$ . What is the measure, in degrees, of the exterior angle adjacent to the sixth interior angle?

60^\circ

80^\circ

120^\circ

140^\circ

150^\circ

Cevabı ve açıklamayı göster

Cevap: $120^\circ$

Cevap

The measure of the exterior angle adjacent to the sixth interior angle is

120^\circ

.

The sum of the interior angles of a 6-sided polygon (hexagon) is given by

(6 - 2) \times 180^\circ = 720^\circ

. The sum of the five given interior angles is

115^\circ + 125^\circ + 130^\circ + 140^\circ + 150^\circ = 660^\circ

, which leaves

720^\circ - 660^\circ = 60^\circ

for the sixth interior angle. Since an interior angle and its adjacent exterior angle are supplementary, the exterior angle is

180^\circ - 60^\circ = 120^\circ

.

Adım Adım Çözüm

1

Calculate the sum of the interior angles for a convex hexagon.

Using

(n - 2) \times 180^\circ

with

n = 6

, the total interior angle sum is

(6 - 2) \times 180^\circ = 4 \times 180^\circ = 720^\circ

.

The sum of interior angles of any convex polygon with

n

sides is

(n - 2) \times 180^\circ

.

2

Find the sum of the five given interior angles.

115^\circ + 125^\circ + 130^\circ + 140^\circ + 150^\circ = 660^\circ

.

Summing the five known angle values is necessary to find the remaining sixth angle.

3

Determine the measure of the sixth interior angle.

720^\circ - 660^\circ = 60^\circ

.

Subtracting the sum of the five interior angles from the total interior angle sum yields the sixth interior angle.

4

Calculate the supplementary exterior angle.

180^\circ - 60^\circ = 120^\circ

.

An interior angle and its adjacent exterior angle form a straight line and are supplementary (

180^\circ

).

Anahtar Kavram

Interior and Exterior Angles of Convex Polygons

Quadrilaterals and Polygons

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Alternatif Yöntem

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Alternatif Yöntem

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap