Plane Geometry Alıştırma Soruları — ACT — Sayfa 4

Soru 61Soru →

In $\triangle PQR$ , the measure of $\angle P$ is $50^\circ$ . The angle bisectors of $\angle PQR$ and $\angle PRQ$ intersect at point $I$ inside the triangle. What is the measure of $\angle QIR$ ?

65^\circ

100^\circ

115^\circ

125^\circ

130^\circ

Cevabı ve açıklamayı göster

Cevap: $115^\circ$

Cevap

115 degrees

To find the measure of the angle at the intersection, we first use the Triangle Angle-Sum Theorem on the larger triangle. The sum of the interior angles in the larger triangle is 180 degrees. Since the angle at one vertex is 50 degrees, the sum of the other two angles must be 130 degrees. The angle bisectors divide these two angles in half, meaning the sum of the two half-angles in the smaller triangle is half of 130 degrees, which is 65 degrees. Applying the Triangle Angle-Sum Theorem to the smaller triangle, the sum of its interior angles is also 180 degrees. Subtracting the sum of the two half-angles (65 degrees) from 180 degrees gives 115 degrees for the angle at the intersection.

Adım Adım Çözüm

1

Find the sum of the remaining interior angles of the triangle.

The sum of the angles at Q and R is 130 degrees.

The sum of all interior angles in any triangle is always 180 degrees, and the angle at P is given as 50 degrees.

2

Determine the sum of the bisected angles in the smaller triangle.

The sum of the half-angles is 65 degrees.

Since the lines QI and RI are angle bisectors, the sum of the interior angles of the smaller triangle at vertices Q and R is half the sum of the angles at Q and R of the larger triangle.

3

Calculate the measure of the angle at the intersection point.

The angle at the intersection point is 115 degrees.

The sum of the interior angles in the smaller triangle is also 180 degrees, so the unknown angle is found by subtracting the sum of the two half-angles from 180 degrees.

Anahtar Kavram

The Triangle Angle-Sum Theorem states that the sum of the measures of the interior angles of a triangle is always 180 degrees. Angle bisectors divide an angle into two equal parts.

Alternatif Yöntem

Alternatively, one can assign specific values to the angles that satisfy the given conditions. Since the sum of the angles in the large triangle must be 180 degrees and the angle at vertex P is 50 degrees, the sum of the other two angles must be 130 degrees. If we assume the triangle is isosceles with the two unknown angles being equal, each of those angles is 65 degrees. Their bisectors would each form an angle of 32.5 degrees with the base. In the smaller triangle, the sum of these two bisected angles is 65 degrees, which leaves 115 degrees for the angle at the intersection point.

Tahmini Süre:1m 15s

Soru 62Soru →

In parallelogram $ABCD$ , the measure of interior angle $A$ is $72^\circ$ . What is the measure, in degrees, of interior angle $B$ ?

Cevabı ve açıklamayı göster

Cevap: 108

Cevap

The measure of interior angle

B

is

108

degrees.

In parallelogram

ABCD

, interior angles

A

and

B

are consecutive angles. A fundamental property of parallelograms is that consecutive angles are supplementary (their measures sum to

180^\circ

). Therefore, the measure of angle

B

is calculated as

180^\circ - 72^\circ = 108^\circ

.

Adım Adım Çözüm

1

Use the consecutive angles property of parallelograms.

The sum of consecutive interior angles

A

and

B

is

180^\circ

.

Since opposite sides of a parallelogram are parallel, consecutive interior angles are supplementary.

2

Set up the equation and solve for the unknown angle.

m\angle B = 108^\circ

Subtract

72^\circ

from

180^\circ

.

Anahtar Kavram

Properties of parallelograms (consecutive angles are supplementary)

Soru 63Soru →

In the right trapezoid $ABCD$ below, $AB$ is parallel to $CD$ , and the measures of $\angle A$ and $\angle D$ are both $90^\circ$ . The length of $CD$ is $7$ , the length of $BC$ is $8$ , and the measure of $\angle B$ is $60^\circ$ . What is the length of the diagonal $BD$ ?

Cevabı ve açıklamayı göster

Cevap: 13

Cevap

13

By drawing altitude

CE

perpendicular to

AB

, we form rectangle

AECD

and right triangle

\triangle CEB

. Since

\angle B = 60^\circ

,

\triangle CEB

is a

30^\circ

-

60^\circ

-

90^\circ

triangle with hypotenuse

BC = 8

. The side opposite

30^\circ

is

BE = 8/2 = 4

, and the side opposite

60^\circ

is

CE = 4\sqrt{3}

. Since opposite sides of rectangle

AECD

are equal, we find

AD = CE = 4\sqrt{3}

and

AE = CD = 7

. Thus,

AB = AE + BE = 7 + 4 = 11

. Finally, we apply the Pythagorean Theorem to right triangle

\triangle DAB

:

BD^2 = AD^2 + AB^2 = (4\sqrt{3})^2 + 11^2 = 48 + 121 = 169

, yielding

BD = 13

.

Adım Adım Çözüm

1

Decompose the trapezoid by drawing an altitude from

C

perpendicular to

AB

, meeting it at

E

.

This forms a rectangle

AECD

and a right triangle

CEB

.

Decomposing the figure allows us to use right triangle trigonometry and parallel line relationships to determine missing side lengths.

2

Calculate the lengths of the legs of right triangle

CEB

.

BE = 4

and

CE = 4\sqrt{3}

.

Since

\angle B = 60^\circ

,

\triangle CEB

is a

30^\circ

-

60^\circ

-

90^\circ

triangle. The shorter leg

BE

is half the hypotenuse

BC

, and the longer leg

CE

is the shorter leg times

\sqrt{3}

.

3

Determine the lengths of

AD

and

AB

.

AD = 4\sqrt{3}

and

AB = 11

.

In the rectangle

AECD

, opposite sides are equal, so

AD = CE = 4\sqrt{3}

and

AE = CD = 7

. Thus, the base

AB = AE + BE = 7 + 4 = 11

.

4

Use the Pythagorean Theorem in right triangle

DAB

to solve for

BD

.

BD = 13

.

In right triangle

DAB

, the legs are

AD = 4\sqrt{3}

and

AB = 11

. The hypotenuse

BD

satisfies

BD^2 = AD^2 + AB^2 = (4\sqrt{3})^2 + 11^2 = 48 + 121 = 169

, so

BD = \sqrt{169} = 13

.

Anahtar Kavram

Solving multi-step geometry problems by decomposing shapes into rectangles and special right triangles (

30^\circ

-

60^\circ

-

90^\circ

), then applying the Pythagorean Theorem.

Soru 64Soru →

A convex hexagon has four interior angles that measure $100^\circ$ , $115^\circ$ , $125^\circ$ , and $140^\circ$ . The remaining two interior angles are in the ratio $3:5$ . What is the degree measure of the largest interior angle of this hexagon?

90^\circ

140^\circ

144^\circ

150^\circ

160^\circ

Cevabı ve açıklamayı göster

Cevap: $150^\circ$

Cevap

The degree measure of the largest interior angle is

150^\circ

.

The sum of the interior angles of a hexagon is

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

. Subtracting the four given angles (

100^\circ

,

115^\circ

,

125^\circ

, and

140^\circ

) from

720^\circ

leaves a remaining sum of

240^\circ

. Since the two remaining angles are in the ratio

3:5

, they can be written as

3x

and

5x

, where

3x + 5x = 240^\circ

. Solving for

x

gives

8x = 240^\circ \implies x = 30^\circ

. The two remaining angles are

90^\circ

and

150^\circ

. Comparing all six interior angles of the hexagon, the largest measure is

150^\circ

.

Adım Adım Çözüm

1

Calculate the sum of the interior angles of a hexagon.

The sum is

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

.

The sum of the interior angles of any convex

n

-sided polygon is given by

(n-2) \times 180^\circ

.

2

Find the sum of the four given interior angles.

100^\circ + 115^\circ + 125^\circ + 140^\circ = 480^\circ

.

This determines the total measure of the known angles.

3

Determine the remaining sum of the two unknown angles.

720^\circ - 480^\circ = 240^\circ

.

Subtracting the sum of the known angles from the total sum yields the combined measure of the remaining two angles.

4

Set up an equation using the ratio

3:5

to find the measures of the two remaining angles.

3x + 5x = 240^\circ \implies 8x = 240^\circ \implies x = 30^\circ

. The two angles are

3(30^\circ) = 90^\circ

and

5(30^\circ) = 150^\circ

.

Representing the two angles in terms of a common variable

x

allows us to solve for their individual measures.

5

Identify the largest interior angle among all six angles of the hexagon.

The largest angle is

150^\circ

.

Comparing all six angles (

90^\circ

,

100^\circ

,

115^\circ

,

125^\circ

,

140^\circ

, and

150^\circ

), the maximum value is

150^\circ

.

Anahtar Kavram

Calculating the interior angle sum of a polygon and solving for unknown angles using ratios.

Soru 65Soru →

An irregular convex hexagon has three interior angles that are congruent to each other, and the remaining three interior angles have measures in the ratio $3:4:5$ . If the sum of the measures of the three congruent angles is $360^\circ$ , what is the degree measure of the largest interior angle of the hexagon?

90^\circ

120^\circ

150^\circ

250^\circ

300^\circ

Cevabı ve açıklamayı göster

Cevap: $150^\circ$

Cevap

The degree measure of the largest interior angle of the hexagon is

150^\circ

.

The correct answer is

150^\circ

. The total sum of the interior angles of a hexagon is

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

. Subtracting the sum of the three congruent angles (

360^\circ

) leaves

360^\circ

for the remaining three angles. With their measures in the ratio

3:4:5

, we can write the equation

3x + 4x + 5x = 360^\circ

, which simplifies to

12x = 360^\circ

, giving

x = 30^\circ

. The measures of these three angles are

90^\circ

,

120^\circ

, and

150^\circ

. The three congruent angles each measure

360^\circ / 3 = 120^\circ

. Comparing all six angles (

120^\circ, 120^\circ, 120^\circ, 90^\circ, 120^\circ, 150^\circ

), the largest measure is

150^\circ

.

Adım Adım Çözüm

1

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

720^\circ

Using the formula

(n - 2) \times 180^\circ

for a polygon with

n = 6

sides, the sum is

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

.

2

Determine the sum of the remaining three interior angles.

360^\circ

We subtract the sum of the three congruent angles (

360^\circ

) from the total sum of the hexagon's interior angles (

720^\circ

):

720^\circ - 360^\circ = 360^\circ

.

3

Set up and solve an equation using the ratio of the remaining three angles.

x = 30^\circ

Let the measures of the remaining three angles be

3x

,

4x

, and

5x

. Their sum is

360^\circ

, so

3x + 4x + 5x = 360^\circ \implies 12x = 360^\circ \implies x = 30^\circ

.

4

Calculate the measures of all six interior angles to identify the largest one.

The angles are

120^\circ

,

120^\circ

,

120^\circ

,

90^\circ

,

120^\circ

, and

150^\circ

. The largest is

150^\circ

.

Each of the three congruent angles measures

360^\circ / 3 = 120^\circ

. The other three angles measure

3(30^\circ) = 90^\circ

,

4(30^\circ) = 120^\circ

, and

5(30^\circ) = 150^\circ

. Comparing all these values, the maximum is

150^\circ

.

Anahtar Kavram

The sum of the interior angles of a convex

n

-sided polygon is given by

(n-2) \times 180^\circ

. Irregular polygons share this total sum, and ratio relationships can be solved by introducing a variable multiplier.

Soru 66Soru →

The measures of five of the interior angles of a convex hexagon are in the ratio $3:4:5:6:7$ . The measure of the sixth interior angle is $30^\circ$ less than the average measure of the other five angles. What is the degree measure of the largest interior angle of the hexagon?

125^\circ

161^\circ

175^\circ

202^\circ

203^\circ

Cevabı ve açıklamayı göster

Cevap: $175^\circ$

Cevap

The correct answer is 175^\circ because solving the equation for the sum of the hexagon's interior angles yields a multiplier of

x = 25

, making the largest angle

7x = 175^\circ

.

The correct answer is 175° because the sum of the interior angles of a hexagon is

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

. Representing the five ratio-based angles as

3x

,

4x

,

5x

,

6x

, and

7x

gives their sum as

25x

and their average as

5x

. The sixth angle is therefore

5x - 30

. Setting up the sum of all six angles yields

25x + (5x - 30) = 720 \implies 30x = 750 \implies x = 25

. The largest angle is

7x = 7(25) = 175^\circ

.

Adım Adım Çözüm

1

Find the sum of the interior angles of a hexagon.

Sum =

720^\circ

Using the interior angle sum formula

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

for a hexagon where

n = 6

, we get

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

.

2

Express the five ratio-based angles in terms of a variable

x

.

The angles are

3x

,

4x

,

5x

,

6x

, and

7x

.

Since the measures of five angles are in the ratio

3:4:5:6:7

, we can define them as multiples of a common scale factor

x

.

3

Compute the average measure of these five angles in terms of

x

.

Average =

5x

The sum of the five angles is

3x + 4x + 5x + 6x + 7x = 25x

. The average is the sum divided by the count:

\frac{25x}{5} = 5x

.

4

Express the measure of the sixth angle in terms of

x

.

Sixth angle =

5x - 30

The sixth angle is described as being

30^\circ

less than the average of the other five angles, which we found to be

5x

.

5

Write and solve the equation for the sum of all six interior angles.

x = 25

The sum of all six angles must equal the total sum of

720^\circ

:

25x + (5x - 30) = 720 \implies 30x - 30 = 720 \implies 30x = 750 \implies x = 25

.

6

Calculate the measure of the largest interior angle.

175^\circ

The largest interior angle corresponds to the largest term in the ratio, which is

7x

. Substituting

x = 25

yields

7 \times 25 = 175^\circ

.

Anahtar Kavram

Applying the interior angle sum formula for polygons combined with ratio and algebraic translation properties.

Alternatif Yöntem

Instead of using the sum of interior angles, one could use the sum of exterior angles, which is always

360^\circ

. The exterior angles corresponding to the five angles in ratio

3:4:5:6:7

would be

180 - 3x

,

180 - 4x

,

180 - 5x

,

180 - 6x

, and

180 - 7x

. The sixth exterior angle is

180 - (5x - 30) = 210 - 5x

. Summing these six exterior angles:

(180 - 3x) + (180 - 4x) + (180 - 5x) + (180 - 6x) + (180 - 7x) + (210 - 5x) = 1110 - 30x = 360

. Solving for

x

yields

30x = 750 \implies x = 25

. The largest interior angle corresponds to the smallest exterior angle, which is

180 - 7x = 180 - 7(25) = 5^\circ

, giving the largest interior angle as

180 - 5 = 175^\circ

.

Tahmini Süre:3m 0s

Soru 67Soru →

In rectangle $PQRS$ , the length of side $PQ$ is 8 inches and the length of diagonal $PR$ is 10 inches. What is the length, in inches, of side $QR$ ?

2

6

6.4

12.8

18

Cevabı ve açıklamayı göster

Cevap: 6

Cevap

The length of side QR is 6 inches.

The correct answer is 6. In a rectangle, all four interior angles are right angles (90 degrees). Therefore, triangle PQR is a right triangle with the right angle at Q. According to the Pythagorean theorem, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse: PQ² + QR² = PR². Substituting the given values: 8² + QR² = 10², which simplifies to 64 + QR² = 100. Subtracting 64 from both sides gives QR² = 36. Taking the square root of both sides yields QR = 6 inches.

Adım Adım Çözüm

1

Identify the geometric relationship in rectangle PQRS.

Angle PQR is a right angle (90 degrees), which makes triangle PQR a right triangle with legs PQ and QR, and hypotenuse PR.

By definition, all interior angles of a rectangle are 90 degrees, and the diagonal connects opposite vertices, forming two right triangles.

2

Apply the Pythagorean theorem to triangle PQR.

PQ² + QR² = PR², which becomes 8² + QR² = 10².

The Pythagorean theorem states that in any right triangle, the sum of the squares of the leg lengths equals the square of the hypotenuse length.

3

Solve for the unknown side length QR.

64 + QR² = 100, which gives QR² = 36, and thus QR = 6.

Subtract 64 from both sides to isolate QR², then take the square root of 36.

Anahtar Kavram

Properties of rectangles and the application of the Pythagorean theorem to right triangles formed by diagonals.

Soru 68Soru →

In $\triangle ABC$ , the measure of exterior angle $\angle ACD$ is $135^\circ$ , where $D$ lies on the extension of side $BC$ past $C$ . If the measure of interior angle $\angle A$ is $25^\circ$ greater than the measure of interior angle $\angle B$ , what is the measure, in degrees, of $\angle B$ ?

Cevabı ve açıklamayı göster

Cevap: 55

Cevap

55

According to the Exterior Angle Theorem, the measure of exterior angle

\angle ACD

is equal to the sum of the measures of its remote interior angles,

\angle A

and

\angle B

. This gives the equation

\text{m}\angle A + \text{m}\angle B = 135^\circ

. Using the information that

\text{m}\angle A = \text{m}\angle B + 25^\circ

, we substitute this expression into the equation to get

(\text{m}\angle B + 25^\circ) + \text{m}\angle B = 135^\circ

. Simplifying this equation gives

2\text{m}\angle B + 25 = 135

, which simplifies to

2\text{m}\angle B = 110

, and dividing by 2 yields

\text{m}\angle B = 55^\circ

.

Adım Adım Çözüm

1

Apply the Exterior Angle Theorem to express the relation between the exterior angle and the two remote interior angles.

\text{m}\angle A + \text{m}\angle B = 135^\circ

The measure of an exterior angle of a triangle is equal to the sum of the measures of its two remote interior angles.

2

Substitute the relationship between the interior angles into the equation.

(\text{m}\angle B + 25^\circ) + \text{m}\angle B = 135^\circ

The problem states that the measure of interior angle

\angle A

is

25^\circ

greater than the measure of interior angle

\angle B

.

3

Solve the algebraic equation for the measure of interior angle

\angle B

.

\text{m}\angle B = 55^\circ

Combining like terms gives

2\text{m}\angle B + 25 = 135

. Subtracting 25 from both sides gives

2\text{m}\angle B = 110

. Dividing by 2 yields

\text{m}\angle B = 55^\circ

.

Anahtar Kavram

Exterior Angle Theorem and remote interior angles relation

Soru 69Soru →

A convex polygon has $n$ sides. The sum of the measures of its interior angles is $3$ times the sum of the measures of its exterior angles. What is the value of $n$ ?

Cevabı ve açıklamayı göster

Cevap: 8

Cevap

8

The sum of the interior angle measures of an

n

-sided convex polygon is

(n-2) \times 180^\circ

. The sum of the exterior angle measures is always

360^\circ

. According to the problem, the sum of the interior angles is

3

times the sum of the exterior angles, which can be written as the equation

(n-2) \times 180 = 3 \times 360

. Simplifying the right side gives

(n-2) \times 180 = 1080

. Dividing both sides by

180

results in

n - 2 = 6

. Adding

2

to both sides yields

n = 8

.

Adım Adım Çözüm

1

Use the formula for the sum of the interior angle measures of a convex polygon with

n

sides.

The sum of the interior angles is

(n-2) \times 180^\circ

.

By the polygon interior angle sum theorem, the sum of the interior angles of any convex

n

-gon is

(n-2) \times 180^\circ

.

2

Identify the sum of the exterior angle measures of a convex polygon.

The sum of the exterior angles is

360^\circ

.

The sum of the exterior angles of any convex polygon is always constant and equals

360^\circ

, regardless of the number of sides.

3

Set up the equation relating the two sums as given in the problem statement.

(n-2) \times 180^\circ = 3 \times 360^\circ

The problem states that the sum of the interior angles is

3

times the sum of the exterior angles.

4

Solve the equation for

n

.

n = 8

Divide both sides by

180^\circ

to get

n - 2 = 6

, then add

2

to both sides to find

n = 8

.

Anahtar Kavram

The sum of the interior angles of a convex polygon with

n

sides is

(n-2) \times 180^\circ

, and the sum of its exterior angles is

360^\circ

.

Soru 70Soru →

The sum of the measures of the interior angles of a convex polygon is $900^\circ$ . How many sides does this polygon have?

3

5

6

7

8

Cevabı ve açıklamayı göster

Cevap: 7

Cevap

The polygon has 7 sides.

The correct answer is

7

because the sum of the interior angles of a convex polygon with

n

sides is

(n - 2) \times 180^\circ

. Setting this equal to

900^\circ

yields

(n - 2) \times 180 = 900

. Dividing both sides by

180

gives

n - 2 = 5

, and adding

2

to both sides results in

n = 7

.

Adım Adım Çözüm

1

Set up the equation for the sum of the interior angles of a polygon.

(n - 2) \times 180^\circ = 900^\circ

The sum of the interior angles of a convex polygon with

n

sides is always

(n - 2) \times 180^\circ

.

2

Divide both sides of the equation by

180^\circ

.

n - 2 = 5

To isolate the term with

n

, we divide

900

by

180

.

3

Solve for

n

by adding

2

to both sides.

n = 7

Adding

2

to both sides isolates the variable

n

.

Anahtar Kavram

The sum of the interior angles of an

n

-sided convex polygon is

(n - 2) \times 180^\circ

.

Tahmini Süre:45s

Soru 71Soru →

A triangle has two sides of length $9$ and $14$ . The length of the third side, $s$ , is a multiple of $4$ . What is the sum of all possible integer values for $s$ ?

48

60

56

76

80

Cevabı ve açıklamayı göster

Cevap: 56

Cevap

56

According to the Triangle Inequality Theorem, the length of the third side,

s

, of a triangle with sides of length 9 and 14 must satisfy the inequality

14 - 9 < s < 14 + 9

, which simplifies to

5 < s < 23

. The integer values within this range that are multiples of 4 are 8, 12, 16, and 20. Adding these values together yields a sum of 56.

Adım Adım Çözüm

1

Apply the Triangle Inequality Theorem to determine the range of possible lengths for the third side.

The third side length,

s

, must satisfy the inequality:

14 - 9 < s < 14 + 9

, which simplifies to

5 < s < 23

.

The Triangle Inequality Theorem states that the length of any side of a triangle must be strictly greater than the positive difference of the other two sides and strictly less than the sum of the other two sides.

2

Identify all integer values in the range

(5, 23)

that are multiples of 4.

The multiples of 4 that are strictly greater than 5 and strictly less than 23 are: 8, 12, 16, and 20.

We must find the integers within the bounds that can be divided by 4 with a remainder of 0.

3

Calculate the sum of the identified multiples of 4.

8 + 12 + 16 + 20 = 56

.

The question asks for the sum of all possible integer values for the third side length

s

.

Anahtar Kavram

Triangle Inequality Theorem

Tahmini Süre:1m 0s

Soru 72Soru →

A regular decagon has $10$ sides of equal length. What is the measure, in degrees, of one exterior angle of this decagon?

Cevabı ve açıklamayı göster

Cevap: 36

Cevap

The correct answer is 36.

The sum of the exterior angles of any convex polygon is always

360^\circ

. A regular decagon has

10

congruent sides and therefore

10

congruent exterior angles. Dividing

360^\circ

by

10

yields

36^\circ

for each exterior angle.

Adım Adım Çözüm

1

Determine the number of exterior angles in a regular decagon.

A decagon has 10 sides, so it has 10 exterior angles.

A polygon has the same number of exterior angles as its number of sides.

2

State the sum of the exterior angles for a convex polygon.

The sum of the exterior angles is 360 degrees.

The exterior angles of any convex polygon always sum to 360 degrees regardless of the number of sides.

3

Calculate the measure of one exterior angle.

360 / 10 = 36

Since the decagon is regular, all of its exterior angles are equal in measure, so we divide the total sum by the number of angles.

Anahtar Kavram

The sum of the exterior angles of any convex polygon is

360^\circ

. For a regular polygon with

n

sides, the measure of each exterior angle is

\frac{360^\circ}{n}

.

Soru 73Soru →

The measures of the interior angles of a convex pentagon are in the ratio $2:3:4:4:5$ . What is the degree measure of the smallest interior angle in this pentagon?

40^\circ

60^\circ

80^\circ

108^\circ

150^\circ

Cevabı ve açıklamayı göster

Cevap: $60^\circ$

Cevap

60^\circ

The sum of the interior angles of a convex pentagon is calculated as

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

. Representing the five angle measures in the given ratio as

2x

,

3x

,

4x

,

4x

, and

5x

, their sum is

18x = 540^\circ

, which gives

x = 30^\circ

. The smallest angle corresponds to the smallest term in the ratio, which is

2x = 2 \times 30^\circ = 60^\circ

. Therefore, the correct measure is

60^\circ

.

Adım Adım Çözüm

1

Determine the sum of the interior angles of a convex pentagon.

The sum is

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

.

The sum of the interior angles of any convex

n

-gon is given by the formula

(n - 2) \times 180^\circ

, and a pentagon has

5

sides.

2

Set up an algebraic equation using the given ratio of the angle measures.

Let the angle measures be

2x

,

3x

,

4x

,

4x

, and

5x

. Their sum is

2x + 3x + 4x + 4x + 5x = 18x = 540^\circ

.

The sum of the actual angle measures must equal the total interior angle sum of the pentagon.

3

Solve for the variable

x

.

x = 540^\circ / 18 = 30^\circ

.

Dividing the total sum by the sum of the ratio parts gives the value of a single ratio unit.

4

Calculate the measure of the smallest interior angle.

Smallest angle

= 2x = 2 \times 30^\circ = 60^\circ

.

The smallest term in the ratio is

2

, so multiplying this term by

x

gives the smallest angle measure.

Anahtar Kavram

The sum of the interior angles of a convex polygon with

n

sides is

(n-2) \times 180^\circ

. For an irregular polygon with angles in a given ratio, the individual angle measures can be found by setting up a linear equation where the sum of the ratio parts multiplied by a variable equals the total sum.

Soru 74Soru →

Is the following statement true or false? In any rectangle, the diagonals must be perpendicular to each other.

Doğru

Yanlış

Cevabı ve açıklamayı göster

Cevap: False

Cevap

False

The correct answer is False because the diagonals of a rectangle are perpendicular only when the rectangle has equal side lengths (making it a square). In a standard rectangle, the diagonals are congruent and bisect each other, but they do not intersect at right angles.

Adım Adım Çözüm

1

Analyze the properties of a general rectangle.

A rectangle is a parallelogram with four right angles. Its diagonals are congruent and bisect each other.

To determine diagonal behavior, we must start with the standard properties of the shape.

2

Examine if the diagonals are perpendicular in all cases.

The diagonals of a rectangle are perpendicular only if the rectangle is a square (all sides equal). For a general rectangle with unequal adjacent sides, the diagonals are not perpendicular.

A single counterexample, such as a rectangle with unequal adjacent sides, shows that the property does not hold universally.

Anahtar Kavram

Properties of rectangle diagonals

Tahmini Süre:45s

Soru 75Soru →

In the standard $(x, y)$ coordinate plane, a circle is centered at the origin $(0,0)$ and has a radius of $8$ . A horizontal chord $AB$ lies entirely in the first and second quadrants at a distance of $4$ units from the $x$ -axis. A point $P$ is located on the circle such that $\triangle ABP$ is a right triangle. If the hypotenuse of $\triangle ABP$ is a diameter of the circle, what is the area of $\triangle ABP$ ?

16\sqrt{3}

32

32\sqrt{3}

32\sqrt{5}

48\sqrt{3}

Cevabı ve açıklamayı göster

Cevap: $32\sqrt{3}$

Cevap

The area of the right triangle is

32\sqrt{3}

.

The correct answer is

32\sqrt{3}

. The horizontal chord

AB

has

y

-coordinate

4

, and its endpoints lie on the circle

x^2 + y^2 = 64

. Solving for

x

gives

x = \pm 4\sqrt{3}

, so the length of the chord is

8\sqrt{3}

. Because the triangle is inscribed in the circle and is a right triangle, its hypotenuse must be a diameter of the circle (length

16

). Since

AB < 16

,

AB

is a leg, and the hypotenuse is one of the other sides (e.g.,

AP

). The remaining leg

BP

is found using the Pythagorean theorem:

BP = \sqrt{16^2 - (8\sqrt{3})^2} = \sqrt{256 - 192} = \sqrt{64} = 8

. The area of the right triangle is

\frac{1}{2} \times 8\sqrt{3} \times 8 = 32\sqrt{3}

.

Adım Adım Çözüm

1

Determine the length of chord

AB

.

The length of chord

AB

is

8\sqrt{3}

.

The equation of the circle is

x^2 + y^2 = 64

. Since the chord is horizontal and at a distance of

4

units from the

x

-axis, its

y

-coordinate is

4

. Substituting

y = 4

gives

x^2 + 16 = 64 \Rightarrow x^2 = 48 \Rightarrow x = \pm 4\sqrt{3}

. The distance between

A(-4\sqrt{3}, 4)

and

B(4\sqrt{3}, 4)

is

8\sqrt{3}

.

2

Apply the rule for a right triangle inscribed in a circle to identify the hypotenuse.

The hypotenuse must be a diameter of length

16

, so the right angle is at

B

(or

A

).

Any right triangle inscribed in a circle must have a diameter as its hypotenuse. The diameter of this circle is

2 \times 8 = 16

. Since the chord

AB = 8\sqrt{3} \approx 13.86

is shorter than the diameter, it cannot be the hypotenuse. Therefore, either

AP

or

BP

is the hypotenuse (a diameter), making the angle opposite to it (either

\angle ABP

or

\angle BAP

) the

90^\circ

angle.

3

Calculate the length of the remaining leg of the right triangle.

The length of leg

BP

is

8

.

Using the Pythagorean theorem for right triangle

ABP

with hypotenuse

AP = 16

and leg

AB = 8\sqrt{3}

:

AB^2 + BP^2 = AP^2 \Rightarrow (8\sqrt{3})^2 + BP^2 = 16^2 \Rightarrow 192 + BP^2 = 256 \Rightarrow BP^2 = 64 \Rightarrow BP = 8

.

4

Compute the area of right triangle

ABP

.

The area is

32\sqrt{3}

.

The area of a right triangle is

\frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times AB \times BP = \frac{1}{2} \times 8\sqrt{3} \times 8 = 32\sqrt{3}

.

Anahtar Kavram

Applying the Pythagorean theorem and Thales's theorem (inscribed right triangles) to solve multi-step geometric problems on the coordinate plane.

Soru 76Soru →

A kite $WXYZ$ has diagonals $WY$ and $XZ$ that intersect at point $P$ . If the length of $WP$ is $4$ centimeters, the length of $PY$ is $9$ centimeters, and the length of $XP$ is $3$ centimeters, what is the measure, in degrees, of angle $WPX$ ?

30

45

60

90

180

Cevabı ve açıklamayı göster

Cevap: 90

Cevap

90 degrees

The diagonals of a kite are always perpendicular to each other. Therefore, the angle formed at their intersection, angle

WPX

, is a right angle, which measures exactly

90

degrees. The given segment lengths are extra information.

Adım Adım Çözüm

1

Identify the fundamental property of the diagonals of a kite.

The diagonals of any kite are perpendicular to each other.

By geometric definition, the diagonals of a kite intersect at a right angle.

2

Determine the measure of the angle formed by the intersection of the diagonals.

Angle

WPX

is a right angle, which measures exactly

90

degrees.

Since the diagonals are perpendicular, their intersection forms four

90

-degree angles regardless of the lengths of the individual diagonal segments.

Anahtar Kavram

The diagonals of a kite are perpendicular (

90^\circ

).

Soru 77Soru →

An irregular convex polygon has $n$ sides. The measures of its interior angles, in degrees, are all distinct integers. If all of the interior angles are obtuse, what is the maximum possible value of $n$ ?

Cevabı ve açıklamayı göster

Cevap: 26

Cevap

The maximum possible value of

n

is 26.

For a convex

26

-gon, the sum of the interior angles is

24 \times 180^\circ = 4320^\circ

. We can choose

26

distinct integer angles from the range

[91^\circ, 179^\circ]

that sum to exactly

4320^\circ

because the maximum possible sum of

26

distinct integers in this range is

4329^\circ

, which is greater than

4320^\circ

. For

n = 27

, the sum of the interior angles must be

25 \times 180^\circ = 4500^\circ

, but the maximum possible sum of

27

distinct integers in the range is only

4482^\circ

, which is less than

4500^\circ

. Therefore,

26

is the maximum value of

n

.

Adım Adım Çözüm

1

Determine the set of possible angle measures.

The angles must be integers in the range

[91^\circ, 179^\circ]

.

Interior angles of a convex polygon must be less than

180^\circ

. Since they are obtuse and distinct integers, they must be strictly greater than

90^\circ

, giving the range

[91, 179]

.

2

Write the sum of the interior angles of a convex

n

-gon.

Sum

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

.

This is the standard formula for the sum of the interior angles of any convex

n

-gon.

3

Find the maximum possible sum of

n

distinct angles in the range

[91, 179]

.

Maximum Sum

= 179n - \frac{n(n - 1)}{2}

.

The maximum sum is achieved by selecting the largest

n

integers from the set:

179, 178, \dots, 179 - (n - 1)

.

4

Set up the inequality and simplify.

n^2 + n - 720 \le 0

.

Since the sum of the angles must be less than or equal to the maximum possible sum, we have

(n-2) \times 180 \le 179n - \frac{n(n-1)}{2}

. Multiplying by 2 and simplifying yields the quadratic inequality.

5

Solve the quadratic inequality for the largest integer

n

.

n = 26

.

Evaluating the quadratic expression for consecutive integers: for

n = 26

,

26^2 + 26 - 720 = -18 \le 0

; for

n = 27

,

27^2 + 27 - 720 = 36 > 0

. Thus, 26 is the maximum possible integer value.

Anahtar Kavram

Sum of interior angles of a convex polygon combined with algebraic optimization.

Soru 78Soru →

If the diagonals of a convex quadrilateral divide the quadrilateral into four triangles of equal perimeter, then the quadrilateral must be a square. Is this statement true or false?

Doğru

Yanlış

Cevabı ve açıklamayı göster

Cevap: False

Cevap

The statement is false because any non-square rhombus satisfies the condition of being divided into four triangles of equal perimeter, yet it is not a square.

The correct answer is false because the equal-perimeter condition only requires the quadrilateral to be a rhombus (having four equal sides), but does not require the interior angles to be 90 degrees. Any non-square rhombus is a valid counterexample.

Adım Adım Çözüm

1

Analyze the given condition of equal perimeters for the four triangles formed by the diagonals of a convex quadrilateral.

The diagonals of any rhombus are perpendicular and bisect each other, dividing the rhombus into four congruent right triangles.

Congruent triangles have identical side lengths, meaning their perimeters are equal. Thus, every rhombus satisfies this property.

2

Determine if all quadrilaterals satisfying this property must be squares.

A square is a regular quadrilateral, meaning it must have both equal side lengths and interior angles of 90 degrees.

To verify if the statement is true, we must test if a non-square rhombus can satisfy the condition.

3

Construct a counterexample using a specific non-square rhombus.

Consider a rhombus with side lengths of 5 units and diagonals of lengths 6 units and 8 units. The diagonals divide it into four right triangles with sides 3, 4, and 5 units. Each triangle has a perimeter of 12 units.

This rhombus has equal perimeters for all four triangles, but its interior angles are not 90 degrees, proving it is not a square.

Anahtar Kavram

The relationship between the diagonals and side properties of rhombuses and squares.

Soru 79Soru →

A convex polygon has $n$ sides. The sum of the measures of its interior angles is $6$ times the sum of the measures of its exterior angles (one at each vertex). What is the value of $n$ ?

Cevabı ve açıklamayı göster

Cevap: 14

Cevap

The number of sides,

n

, of the convex polygon is 14.

The sum of the interior angles of a convex polygon with

n

sides is given by the formula

(n-2) \times 180^\circ

, and the sum of its exterior angles is always

360^\circ

. According to the problem, the sum of the interior angles is

6

times the sum of the exterior angles, yielding the equation

(n - 2) \times 180 = 6 \times 360

. Dividing both sides of the equation by

180

gives

n - 2 = 12

. Adding

2

to both sides results in

n = 14

.

Adım Adım Çözüm

1

State the sum of interior and exterior angles formulas.

Interior sum =

(n-2) \times 180^\circ

, Exterior sum =

360^\circ

.

To represent the geometric properties of the polygon algebraically.

2

Set up the equation based on the given ratio.

(n-2) \times 180 = 6 \times 360

.

The problem states the interior sum is 6 times the exterior sum.

3

Solve the equation for

n

.

n = 14

.

Divide by 180 to get

n - 2 = 12

, then add 2 to both sides.

Anahtar Kavram

The sum of the interior angles of an

n

-sided convex polygon is

(n-2) \times 180^\circ

, and the sum of the exterior angles (one per vertex) is always

360^\circ

.

Soru 80Soru →

An irregular convex hexagon has two interior angles measuring $90^\circ$ and $130^\circ$ , respectively. The remaining four interior angles have measures in the ratio $5:6:7:7$ . What is the degree measure of the largest interior angle in this hexagon?

100^\circ

120^\circ

130^\circ

140^\circ

150^\circ

Cevabı ve açıklamayı göster

Cevap: $140^\circ$

Cevap

The correct answer is

140^\circ

.

The sum of the interior angles of a hexagon is

720^\circ

. Subtracting the two given angles (

90^\circ

and

130^\circ

) leaves

500^\circ

for the remaining four angles. Since these angles are in the ratio

5:6:7:7

, we represent them as

5x

,

6x

,

7x

, and

7x

. Their sum is

25x = 500^\circ

, which gives

x = 20^\circ

. The largest of the remaining angles is

7x = 7(20^\circ) = 140^\circ

. Since

140^\circ

is greater than both

90^\circ

and

130^\circ

, it is the largest interior angle of the hexagon.

Adım Adım Çözüm

1

Calculate the sum of the interior angles of a hexagon.

Using the formula for the sum of the interior angles of a polygon with

n

sides,

(n - 2) \times 180^\circ

, for a hexagon (

n = 6

), the sum is

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

.

To establish the total sum of all interior angles of the polygon.

2

Subtract the two known angle measures from the total sum.

720^\circ - 90^\circ - 130^\circ = 500^\circ

.

To find the sum of the remaining four interior angles.

3

Set up an algebraic equation to find the value of one ratio unit,

x

.

5x + 6x + 7x + 7x = 500^\circ \implies 25x = 500^\circ \implies x = 20^\circ

.

To determine the constant multiplier for the ratio of the remaining angles.

4

Calculate the measures of the remaining angles and determine the largest angle.

The remaining angles are

5(20^\circ) = 100^\circ

,

6(20^\circ) = 120^\circ

,

7(20^\circ) = 140^\circ

, and

7(20^\circ) = 140^\circ

. Comparing all six angles of the hexagon (

90^\circ, 100^\circ, 120^\circ, 130^\circ, 140^\circ, 140^\circ

), the largest angle is

140^\circ

.

To identify the maximum interior angle measure of the hexagon.

Anahtar Kavram

The sum of the interior angles of a convex polygon with

n

sides is

(n - 2) \times 180^\circ

. The individual angle measures in an irregular polygon can be determined using algebraic representations of their relationships or ratios.

Alternatif Yöntem

Once the value of the ratio unit

x = 20^\circ

is determined, you can quickly find the largest candidate angle by multiplying the largest ratio component (

7

) by

x

to get

7(20^\circ) = 140^\circ

, and then compare it to the given angles (

90^\circ

and

130^\circ

) to verify that it is indeed the largest.

Tahmini Süre:1m 30s

Plane Geometry

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