Area of Two-Dimensional Shapes Alıştırma Soruları — SAT

Soru 1Soru →

A sign is in the shape of a parallelogram with a base of $18$ inches and a height of $10$ inches. A smaller, similar parallelogram is cut out from the center of the sign. If the scale factor of the smaller parallelogram to the larger parallelogram is $1$ to $3$ , what is the area, in square inches, of the remaining portion of the sign?

Cevabı ve açıklamayı göster

Cevap: 160

Cevap

160

The area of the larger parallelogram is

18 \times 10 = 180

square inches. Since the smaller parallelogram is similar to the larger one with a scale factor of

1

to

3

, the area of the smaller parallelogram is

\left(\frac{1}{3}\right)^2 = \frac{1}{9}

of the area of the larger parallelogram. This gives an area of

180 \times \frac{1}{9} = 20

square inches for the smaller parallelogram. Subtracting this from the larger area yields

180 - 20 = 160

square inches.

Adım Adım Çözüm

1

Calculate the area of the larger parallelogram.

A_{\text{larger}} = 180

square inches

The area of a parallelogram is the product of its base and height (

A = bh

).

2

Determine the area of the smaller, similar parallelogram.

A_{\text{smaller}} = 20

square inches

The ratio of the areas of similar figures is the square of the scale factor:

\left(\frac{1}{3}\right)^2 = \frac{1}{9}

.

3

Subtract the smaller area from the larger area to find the remaining area.

A_{\text{remaining}} = 160

square inches

The remaining portion of the sign is the total area minus the cutout area.

Anahtar Kavram

The area of similar geometric shapes scales by the square of the linear scale factor.

Soru 2Soru →

A square picture has an area of $16$ square inches. If the side length of the picture is doubled, what is the area, in square inches, of the new picture?

8

32

64

128

Cevabı ve açıklamayı göster

Cevap: $64$

Cevap

64

square inches

The area of a square with side length

s

is given by

s^2

. Since the original area is

16

square inches, the original side length is

\sqrt{16} = 4

inches. Doubling this side length gives a new side length of

8

inches. Therefore, the area of the new square is

8^2 = 64

square inches. Alternatively, when all linear dimensions of a two-dimensional shape are multiplied by a scale factor

k

, the area is multiplied by

k^2

. Since the side length is doubled (

k = 2

), the area is multiplied by

2^2 = 4

, resulting in

16 \times 4 = 64

square inches.

Adım Adım Çözüm

1

Determine the side length of the original square.

The side length is

\sqrt{16} = 4

inches.

The area of a square is given by

A = s^2

, where

s

is the side length. So,

s = \sqrt{A}

.

2

Calculate the side length of the new square after doubling.

The new side length is

4 \times 2 = 8

inches.

The problem states that the side length is doubled.

3

Calculate the area of the new square.

The new area is

8^2 = 64

square inches.

The area of the new square is the square of its new side length (

8 \text{ inches} \times 8 \text{ inches}

).

Anahtar Kavram

When a two-dimensional shape is scaled by a factor of

k

, its area is scaled by a factor of

k^2

.

Soru 3Soru →

A trapezoid has parallel bases of length $6$ centimeters and $10$ centimeters, and a height of $5$ centimeters. What is the area, in square centimeters, of the trapezoid?

Cevabı ve açıklamayı göster

Cevap: 40

Cevap

The area of the trapezoid is

40

square centimeters.

The area of a trapezoid is calculated using the formula

A = \frac{a + b}{2} h

, where

a

and

b

are the lengths of the parallel bases and

h

is the height. Substituting the given values

a = 6

,

b = 10

, and

h = 5

into the formula yields

A = \frac{6 + 10}{2} \times 5 = 8 \times 5 = 40

.

Adım Adım Çözüm

1

Recall the area formula for a trapezoid.

A = \frac{a + b}{2}h

The area of a trapezoid is the average of its parallel bases multiplied by its height.

2

Substitute the base values of

6

and

10

, and the height value of

5

into the formula.

A = \frac{6 + 10}{2} \times 5

This sets up the calculation with the given measurements.

3

Simplify the expression to find the final area.

A = 40

Adding the bases gives

16

, dividing by

2

gives

8

, and multiplying by the height of

5

results in

40

.

Anahtar Kavram

Area of a Trapezoid

Tahmini Süre:45s

Soru 4Soru →

A circular rug has an area of $16\pi$ square feet. If the radius of the rug is tripled, what is the area, in square feet, of the new rug?

32\pi

48\pi

144\pi

64\pi

Cevabı ve açıklamayı göster

Cevap: $144\pi$

Cevap

The area of the new rug is

144\pi

square feet.

The area of a circle scales with the square of its radius. If the radius is tripled, the area is multiplied by

3^2 = 9

. Since the original area is

16\pi

square feet, the area of the new rug is

16\pi \times 9 = 144\pi

square feet.

Adım Adım Çözüm

1

Determine the relationship between the linear scale factor and the area scale factor of a circle.

If the radius of a circle is multiplied by a scale factor

k

, the area of the circle is multiplied by

k^2

.

The area of a circle is proportional to the square of its radius (

A = \pi r^2

).

2

Find the area scale factor when the radius is tripled.

Since the radius is multiplied by

3

, the area scale factor is

3^2 = 9

.

Squaring the linear scale factor of 3 gives the quadratic scale factor for the area.

3

Calculate the area of the new rug.

The new area is

16\pi \times 9 = 144\pi

square feet.

Multiplying the original area of

16\pi

square feet by the area scale factor of 9 yields the final scaled area.

Anahtar Kavram

Area scaling of two-dimensional shapes

Tahmini Süre:1m 0s

Soru 5Soru →

A triangular banner has an area of $30$ square feet. If the height of the banner is $5$ feet, what is the length, in feet, of the base of the banner?

Cevabı ve açıklamayı göster

Cevap: 12

Cevap

The length of the base of the banner is

12

feet.

To find the base of the triangular banner, apply the formula for the area of a triangle:

A = \frac{1}{2} b h

. Substituting

30

for the area

A

and

5

for the height

h

yields

30 = \frac{1}{2} b (5)

, or

30 = 2.5b

. Dividing both sides of the equation by

2.5

gives

b = 12

. Therefore, the length of the base of the banner is

12

feet.

Adım Adım Çözüm

1

Recall the formula for the area of a triangle.

A = \frac{1}{2} b h

The area of a triangle is equal to half the product of its base and height.

2

Substitute the given values into the area formula.

30 = \frac{1}{2} \times b \times 5

The problem provides the area (

30

square feet) and the height (

5

feet).

3

Solve for the base

b

.

b = 12

Multiply both sides of the equation by

2

to clear the fraction, giving

60 = 5b

. Then, divide both sides by

5

to find that

b = 12

.

Anahtar Kavram

Area of a triangle

Soru 6Soru →

In the $xy$ -plane, a square has vertices at $(0,0)$ , $(10,0)$ , $(10,10)$ , and $(0,10)$ . A line that passes through the points $(0,2)$ and $(8,10)$ divides the square into two regions. What is the area of the larger region?

Cevabı ve açıklamayı göster

Cevap: 68

Cevap

The area of the larger region is 68.

The total area of the square is

10^2 = 100

. The line segment between

(0,2)

on the left boundary and

(8,10)

on the top boundary forms a right triangle with the top-left vertex of the square

(0,10)

. The legs of this right triangle have lengths

10 - 2 = 8

and

8 - 0 = 8

, so its area is

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

. The area of the other region is

100 - 32 = 68

. The larger area is therefore 68.

Adım Adım Çözüm

1

Calculate the area of the square

100

The square has vertices at

(0,0)

,

(10,0)

,

(10,10)

, and

(0,10)

, which gives a side length of 10. The area of a square is

\text{side}^2

.

2

Determine the dimensions of the smaller triangular region formed by the line

Legs of length 8 and 8

The line intersects the left edge at

(0,2)

and the top edge at

(8,10)

. The corner of the square is at

(0,10)

. The distance from

(0,2)

to

(0,10)

is 8, and the distance from

(8,10)

to

(0,10)

is 8.

3

Calculate the area of the smaller triangular region

32

The area of a right triangle is

\frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 8 \times 8 = 32

.

4

Subtract the area of the smaller region from the total area of the square to find the area of the larger region

68

The area of the larger region is the total area of the square minus the area of the smaller region:

100 - 32 = 68

.

Anahtar Kavram

Calculating the area of a region by partitioning a geometric shape or using subtraction of areas.

Soru 7Soru →

A landscape architect is designing two similar gardens. The smaller garden is in the shape of a regular hexagon and has an area of $24\sqrt{3}$ square meters. The larger garden is also a regular hexagon, and its perimeter is $3$ times the perimeter of the smaller garden. An inscribed circular fountain is to be placed in the center of the larger garden, touching all six sides. What is the area, in square meters, of the circular fountain?

12\pi

36\pi

108\pi

144\pi

Cevabı ve açıklamayı göster

Cevap: $108\pi$

Cevap

The area of the circular fountain is

108\pi

square meters.

The side length of the smaller hexagon is calculated to be

4

meters using the area formula for a regular hexagon. Because the larger hexagon's perimeter is scaled by a factor of

3

, its side length is

12

meters. The radius of the inscribed circle is the apothem of the larger hexagon, which is

12 \times \frac{\sqrt{3}}{2} = 6\sqrt{3}

meters. Squaring this radius and multiplying by

\pi

results in an area of

108\pi

square meters.

Adım Adım Çözüm

1

Find the side length of the smaller regular hexagon.

The side length is

4

meters.

The area of a regular hexagon with side length

s_1

is given by

A = \frac{3\sqrt{3}}{2} s_1^2

. Setting this equal to

24\sqrt{3}

yields

s_1^2 = 16

, so

s_1 = 4

.

2

Determine the side length of the larger regular hexagon.

The side length is

12

meters.

Since the two hexagons are similar, the ratio of their perimeters is equal to the linear scale factor of their sides. The perimeter of the larger hexagon is

3

times that of the smaller hexagon, so its side length is

s_2 = 3 \times 4 = 12

.

3

Find the radius of the inscribed circle in the larger hexagon.

The radius is

6\sqrt{3}

meters.

The radius

r

of the inscribed circle is the apothem of the regular hexagon. For a hexagon with side length

s_2 = 12

, the apothem is

r = s_2 \frac{\sqrt{3}}{2} = 12 \frac{\sqrt{3}}{2} = 6\sqrt{3}

.

4

Calculate the area of the inscribed circular fountain.

The area is

108\pi

square meters.

Using the area formula for a circle,

A = \pi r^2

, we substitute

r = 6\sqrt{3}

to get

A = \pi (6\sqrt{3})^2 = 108\pi

.

Anahtar Kavram

Area of regular polygons, similarity ratio scaling, and properties of inscribed circles.

Soru 8Soru →

An aerospace company designs two similar solar panels. The smaller panel is in the shape of a parallelogram and has an area of $80$ square centimeters. If the perimeter of the larger panel is $1.5$ times the perimeter of the smaller panel, what is the area, in square centimeters, of the larger panel?

120

200

180

270

Cevabı ve açıklamayı göster

Cevap: 180

Cevap

180

Since the two solar panels are similar, the ratio of their perimeters is equal to their linear scale factor,

k = 1.5

. The ratio of their areas is equal to the square of the linear scale factor,

k^2 = (1.5)^2 = 2.25

. Therefore, the area of the larger panel is the product of the smaller panel's area and the area scale factor:

80 \times 2.25 = 180

square centimeters.

Adım Adım Çözüm

1

Determine the linear scale factor between the two similar solar panels.

The linear scale factor is

k = 1.5

.

For similar figures, the ratio of any corresponding linear measurements (such as perimeters) is equal to the linear scale factor. Here, the perimeter of the larger panel is

1.5

times that of the smaller panel.

2

Calculate the area scale factor by squaring the linear scale factor.

The area scale factor is

k^2 = (1.5)^2 = 2.25

.

The ratio of the areas of two similar two-dimensional shapes is equal to the square of their linear scale factor.

3

Multiply the area of the smaller solar panel by the area scale factor to find the area of the larger panel.

The area of the larger solar panel is

80 \times 2.25 = 180

square centimeters.

Multiplying the original area by the area scale factor yields the scaled area of the larger similar figure.

Anahtar Kavram

Area scaling of similar figures

Soru 9Soru →

In rectangle $ABCD$ , the length of side $AB$ is $12$ and the length of side $BC$ is $9$ . Point $P$ lies on the diagonal $AC$ such that $AP = \frac{1}{3} AC$ . What is the area of triangle $BPD$ ?

Cevabı ve açıklamayı göster

Cevap: 18

Cevap

18

The correct answer is 18. The area of the right triangle

ABD

is half of the area of rectangle

ABCD

, which is

\frac{12 \times 9}{2} = 54

. Since point

P

lies on diagonal

AC

such that

AP = \frac{1}{3} AC

, triangle

ABP

has a base of

AP

along line

AC

and shares vertex

B

with triangle

ABC

. Thus, its area is

\frac{1}{3}

of the area of triangle

ABC

, which is

\frac{54}{3} = 18

. Similarly, triangle

ADP

shares vertex

D

with triangle

ADC

and has base

AP

, so its area is

\frac{1}{3}

of the area of triangle

ADC

, which is

\frac{54}{3} = 18

. Because

AP = \frac{1}{3} AC < \frac{1}{2} AC

, point

P

lies inside triangle

ABD

. Therefore, the area of triangle

BPD

is the area of triangle

ABD

minus the areas of triangles

ABP

and

ADP

, which is

54 - 18 - 18 = 18

.

Adım Adım Çözüm

1

Calculate the area of triangle

ABD

.

\text{Area}(\triangle ABD) = \frac{12 \times 9}{2} = 54

The diagonal

BD

divides the rectangle

ABCD

into two congruent right triangles, each with an area equal to half of the rectangle's total area.

2

Find the areas of triangles

ABP

and

ADP

using the ratio of

AP

to

AC

.

\text{Area}(\triangle ABP) = \frac{1}{3} \times \text{Area}(\triangle ABC) = 18

and

\text{Area}(\triangle ADP) = \frac{1}{3} \times \text{Area}(\triangle ADC) = 18

.

Triangles

ABP

and

ABC

share the same altitude from vertex

B

to the line containing diagonal

AC

. Therefore, the ratio of their areas is equal to the ratio of their bases, which is

\frac{AP}{AC} = \frac{1}{3}

. The same logic applies to triangles

ADP

and

ADC

with vertex

D

.

3

Subtract the areas of triangles

ABP

and

ADP

from the area of triangle

ABD

to find the area of triangle

BPD

.

\text{Area}(\triangle BPD) = 54 - 18 - 18 = 18

Since

AP = \frac{1}{3} AC

, which is less than half the length of the diagonal, point

P

lies strictly within the interior of triangle

ABD

. Thus, the area of triangle

ABD

is partitioned into the areas of triangles

ABP

,

ADP

, and

BPD

.

Anahtar Kavram

Partitioning the area of a polygon and using the ratio of bases for triangles sharing a vertex to compute sub-areas.

Alternatif Yöntem

Alternatively, place the rectangle in a coordinate system with

B

at the origin

(0,0)

,

C

at

(12,0)

,

A

at

(0,9)

, and

D

at

(12,9)

. The coordinates of point

P

on diagonal

AC

(from

(0,9)

to

(12,0)

) at one-third of the distance from

A

to

C

are

x = 0 + \frac{1}{3}(12 - 0) = 4

and

y = 9 + \frac{1}{3}(0 - 9) = 6

. The area of triangle

BPD

with vertices

B(0,0)

,

P(4,6)

, and

D(12,9)

can be found using the shoelace formula:

\text{Area} = \frac{1}{2} |0(6 - 9) + 4(9 - 0) + 12(0 - 6)| = \frac{1}{2} |36 - 72| = 18

.

Tahmini Süre:2m 0s

Soru 10Soru →

A metallic plate is in the shape of a trapezoid. The parallel sides of the plate have lengths of $9$ centimeters and $30$ centimeters. The two non-parallel sides have lengths of $10$ centimeters and $17$ centimeters. What is the area, in square centimeters, of the metallic plate?

Cevabı ve açıklamayı göster

Cevap: 156

Cevap

156

To find the area of the trapezoid, we can find its height by drawing perpendicular lines from the vertices of the shorter base to the longer base. This splits the trapezoid into a rectangle of width

9\text{ cm}

and two right triangles with hypotenuses of

10\text{ cm}

and

17\text{ cm}

. The sum of the bases of these two triangles is

30 - 9 = 21\text{ cm}

. Letting their bases be

x

and

y

(where

x + y = 21

), we apply the Pythagorean theorem:

h^2 + x^2 = 100

and

h^2 + y^2 = 289

. Subtracting the equations yields

y^2 - x^2 = 189

. Since

y^2 - x^2 = (y - x)(y + x)

, we have

21(y - x) = 189

, which simplifies to

y - x = 9

. Solving the system

x + y = 21

and

y - x = 9

yields

x = 6

and

y = 15

. Using the Pythagorean triple

6

-

8

-

10

, the height

h

is

8\text{ cm}

. The area is

\frac{1}{2}(9 + 30)(8) = 156

square centimeters.

Adım Adım Çözüm

1

Drop perpendicular lines (heights) from the endpoints of the shorter base to the longer base.

The trapezoid is decomposed into a central rectangle with a width of

9

centimeters and two right triangles with bases

x

and

y

and hypotenuses of

10

centimeters and

17

centimeters, respectively.

This sets up a system of equations relating the heights and bases of the right triangles.

2

Determine the relationship between the bases of the two right triangles.

The sum of the bases of the two right triangles is

x + y = 30 - 9 = 21

centimeters.

The sum of the bases of the right triangles plus the width of the rectangle equals the total length of the longer base of the trapezoid.

3

Set up equations using the Pythagorean theorem for the two right triangles.

h^2 + x^2 = 100

and

h^2 + y^2 = 289

.

Both right triangles share the same height

h

of the trapezoid.

4

Solve for the difference between the two triangle bases.

Subtracting the first equation from the second gives

y^2 - x^2 = 189

. Factoring yields

(y - x)(y + x) = 189

. Substituting

y + x = 21

yields

21(y - x) = 189 \implies y - x = 9

.

Subtracting the equations eliminates the height variable, allowing us to find the difference between the bases.

5

Solve the system of linear equations for

x

and

y

.

Adding

x + y = 21

and

y - x = 9

gives

2y = 30 \implies y = 15

. Substituting back gives

x = 6

.

This determines the exact base segments of both right triangles.

6

Calculate the height of the trapezoid.

h = \sqrt{10^2 - 6^2} = \sqrt{64} = 8

centimeters.

The height of the trapezoid is required to calculate its area.

7

Compute the area of the trapezoid.

\text{Area} = \frac{1}{2}(9 + 30)(8) = 156

square centimeters.

This uses the standard formula for the area of a trapezoid.

Anahtar Kavram

Decomposing a non-isosceles trapezoid into a rectangle and two right triangles to solve for the height using systems of quadratic equations derived from the Pythagorean theorem.

Tahmini Süre:1m 30s

Soru 11Soru →

An artist designs a logo consisting of a rectangle and an isosceles triangle. The rectangle has a length of $12$ centimeters and a width of $8$ centimeters. The base of the isosceles triangle is one of the $8$ -centimeter sides of the rectangle, and the vertex of the triangle lies outside the rectangle. If the total area of the logo is $120$ square centimeters, what is the height, in centimeters, of the triangle?

3

4

6

12

Cevabı ve açıklamayı göster

Cevap: 6

Cevap

6

The total area of the logo is the sum of the area of the rectangle and the area of the triangle. The area of the rectangle is the product of its length and width:

12 \times 8 = 96

square centimeters. Subtracting the area of the rectangle from the total area gives the area of the triangle:

120 - 96 = 24

square centimeters. The area of a triangle is given by the formula

\frac{1}{2} \times \text{base} \times \text{height}

. The base of the triangle is one of the

8

-centimeter sides of the rectangle. Substituting the base and area into the formula gives

24 = \frac{1}{2} \times 8 \times \text{height}

, which simplifies to

24 = 4 \times \text{height}

. Dividing both sides by

4

yields a height of

6

centimeters.

Adım Adım Çözüm

1

Calculate the area of the rectangle.

The area of the rectangle is

12 \times 8 = 96

square centimeters.

The total area of the logo is composite, so we must find the area of the rectangular portion first.

2

Calculate the area of the triangle.

The area of the triangle is

120 - 96 = 24

square centimeters.

Subtracting the rectangle's area from the total area of the logo yields the remaining area occupied by the triangle.

3

Solve for the height of the triangle using the area formula.

The height of the triangle is

6

centimeters.

The base of the triangle is

8

centimeters. Using the formula

\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}

, we set up the equation

24 = \frac{1}{2} \times 8 \times \text{height}

, which simplifies to

24 = 4 \times \text{height}

, giving a height of

6

centimeters.

Anahtar Kavram

The area of a composite shape is the sum of the areas of its simpler component shapes.

Soru 12Soru →

A digital designer is scaling two similar triangular logos. The smaller logo has a perimeter of $18$ centimeters and an area of $12$ square centimeters. If the larger logo has a perimeter of $54$ centimeters, what is the area, in square centimeters, of the larger logo?

36

72

108

324

Cevabı ve açıklamayı göster

Cevap: $108$

Cevap

108 square centimeters

The correct answer is

108

. Since the two triangular logos are similar, the ratio of their areas is the square of the ratio of their perimeters. The ratio of their perimeters is

\frac{54}{18} = 3

, meaning the larger logo has linear dimensions that are

3

times those of the smaller logo. Squaring this scale factor gives the area scaling factor, which is

3^2 = 9

. Therefore, the area of the larger logo is

12 \times 9 = 108

square centimeters.

Adım Adım Çözüm

1

Determine the linear scale factor between the two similar logos.

The linear scale factor

k

is

\frac{54}{18} = 3

.

For similar figures, the ratio of any corresponding linear dimensions (such as perimeters) is equal to the linear scale factor.

2

Find the area scaling factor.

The area scaling factor is

k^2 = 3^2 = 9

.

The ratio of the areas of two similar figures is equal to the square of their linear scale factor.

3

Calculate the area of the larger logo.

The area of the larger logo is

12 \times 9 = 108

square centimeters.

Multiplying the area of the smaller logo by the area scaling factor yields the area of the larger logo.

Anahtar Kavram

The ratio of the areas of two similar two-dimensional shapes is equal to the square of their linear scale factor.

Tahmini Süre:1m 30s

Soru 13Soru →

A circular tabletop is designed to have a square glass inlay. The square glass inlay is inscribed in the circle and has an area of $64$ square inches. What is the area, in square inches, of the circular tabletop?

16\pi

32\pi

64\pi

128\pi

Cevabı ve açıklamayı göster

Cevap: $32\pi$

Cevap

The area of the circular tabletop is

32\pi

square inches.

The correct answer is

32\pi

because the inscribed square has a side length of

8

inches. The diagonal of the square serves as the diameter of the circle, which is

8\sqrt{2}

inches. The radius is therefore

4\sqrt{2}

inches, and the area of the circle is

\pi (4\sqrt{2})^2 = 32\pi

square inches.

Adım Adım Çözüm

1

Find the side length of the inscribed square from its area.

The side length of the square is

8

inches.

The area of a square is given by

s^2

, where

s

is the side length. Since the area is

64

, we solve

s^2 = 64

to find

s = 8

.

2

Find the length of the diagonal of the square, which represents the diameter of the circle.

The diagonal length is

8\sqrt{2}

inches.

In a square with side length

s

, the diagonal is

s\sqrt{2}

. Since the square is inscribed in the circle, the diagonal of the square is equal to the diameter of the circle.

3

Calculate the radius of the circle and then the area.

The radius is

4\sqrt{2}

inches and the area is

32\pi

square inches.

The radius

r

is half of the diameter, so

r = \frac{8\sqrt{2}}{2} = 4\sqrt{2}

. The area of the circle is given by

A = \pi r^2 = \pi (4\sqrt{2})^2 = 32\pi

.

Anahtar Kavram

Relating the area of an inscribed polygon to the circumscribed circle

Tahmini Süre:1m 30s

Soru 14Soru →

In square $ABCD$ , the side length is $12$ . Point $E$ lies on side $AB$ such that $AE = 3BE$ , and point $F$ lies on side $AD$ such that $AF = FD$ . What is the area of triangle $CEF$ ?

Cevabı ve açıklamayı göster

Cevap: 63

Cevap

63

To find the area of triangle

CEF

, we subtract the areas of the three right triangles surrounding it from the total area of square

ABCD

. The area of square

ABCD

is

12^2 = 144

. Point

E

on side

AB

splits the side of length

12

into segments

AE = 9

and

BE = 3

. Point

F

on side

AD

splits the side of length

12

into equal segments

AF = 6

and

FD = 6

. The areas of the three surrounding right triangles are:

\text{Area}(\triangle AEF) = \frac{1}{2} \times 9 \times 6 = 27

,

\text{Area}(\triangle EBC) = \frac{1}{2} \times 3 \times 12 = 18

, and

\text{Area}(\triangle FDC) = \frac{1}{2} \times 6 \times 12 = 36

. Subtracting these from the total area of the square yields

\text{Area}(\triangle CEF) = 144 - (27 + 18 + 36) = 144 - 81 = 63

.

Adım Adım Çözüm

1

Determine the lengths of the segments created by points

E

and

F

on the sides of the square.

Since the square has a side length of

12

, the length of side

AB

is

12

. Given that

AE = 3BE

and

AE + BE = 12

, we can write

3BE + BE = 12 \implies 4BE = 12 \implies BE = 3

. This gives

AE = 9

. Since

F

is the midpoint of

AD

(

AF = FD

), we have

AF = FD = \frac{12}{2} = 6

.

Finding these segment lengths is necessary to compute the base and height of the right triangles at the corners of the square.

2

Calculate the areas of the three right triangles surrounding triangle

CEF

.

The area of right triangle

AEF

is

\frac{1}{2} \times AE \times AF = \frac{1}{2} \times 9 \times 6 = 27

. The area of right triangle

EBC

is

\frac{1}{2} \times BE \times BC = \frac{1}{2} \times 3 \times 12 = 18

. The area of right triangle

FDC

is

\frac{1}{2} \times FD \times CD = \frac{1}{2} \times 6 \times 12 = 36

.

These three triangles occupy the entire area of the square except for the region defined by triangle

CEF

.

3

Subtract the sum of the areas of the three right triangles from the total area of square

ABCD

.

The total area of square

ABCD

is

12^2 = 144

. The area of triangle

CEF

is

144 - (27 + 18 + 36) = 144 - 81 = 63

.

This subtraction removes the corner regions, leaving only the area of the central triangle.

Anahtar Kavram

Calculating the area of an inscribed polygon by subtracting the areas of simpler surrounding geometric shapes from a larger bounding shape.

Alternatif Yöntem

Alternatively, coordinate geometry can be used. Place the vertex

D

at the origin

(0,0)

on the coordinate plane. Then the coordinates of the vertices of the square are

D(0,0)

,

C(12,0)

,

B(12,12)

, and

A(0,12)

. Point

E

lies on segment

AB

and is located at

(9,12)

. Point

F

lies on segment

AD

and is located at

(0,6)

. The area of the triangle with vertices

C(12,0)

,

E(9,12)

, and

F(0,6)

can be found using the Shoelace Formula:

\text{Area} = \frac{1}{2} |12(12 - 6) + 9(6 - 0) + 0(0 - 12)| = \frac{1}{2} |72 + 54 + 0| = \frac{1}{2} (126) = 63

.

Tahmini Süre:1m 30s

Soru 15Soru →

A landscape architect is designing a courtyard in the shape of a right trapezoid. The parallel sides of the courtyard have lengths of $24$ yards and $40$ yards. The side perpendicular to the parallel sides has a length of $15$ yards. A straight path is built from the midpoint of the longer parallel side to the vertex of the shorter parallel side that is adjacent to the perpendicular side, dividing the courtyard into two regions. What is the area, in square yards, of the smaller region?

Cevabı ve açıklamayı göster

Cevap: 150

Cevap

The area of the smaller region is 150 square yards.

The straight path divides the right trapezoid into two regions: a right triangle and a quadrilateral. The right triangle has a vertical leg of 15 yards (the height of the trapezoid) and a horizontal leg of 20 yards (half of the longer parallel side of 40 yards). The area of this right triangle is 0.5 * 20 * 15 = 150 square yards. The total area of the trapezoid is 0.5 * (24 + 40) * 15 = 480 square yards, making the area of the quadrilateral region 480 - 150 = 330 square yards. Comparing the two regions, the smaller region has an area of 150 square yards.

Adım Adım Çözüm

1

Find the length of the segment from the perpendicular corner to the midpoint of the longer parallel side.

20 yards

The midpoint divides the 40-yard side into two equal parts of 20 yards each.

2

Determine the shape and dimensions of the region containing the perpendicular side.

A right triangle with legs of 15 yards and 20 yards.

Since the path goes from the midpoint of the base to the opposite vertex of the perpendicular height, it forms a right triangle with the height and half of the longer base.

3

Calculate the area of this right triangle.

150 square yards

Using the area formula for a triangle, Area = 0.5 * base * height = 0.5 * 20 * 15 = 150.

4

Calculate the total area of the trapezoid and the area of the remaining region to confirm which is smaller.

Total area is 480 square yards; the other region's area is 330 square yards.

The total area is 0.5 * (24 + 40) * 15 = 480. The other region has an area of 480 - 150 = 330. Comparing 150 and 330, 150 is the smaller area.

Anahtar Kavram

Area of composite shapes and trapezoids

Soru 16Soru →

A circular fountain with a diameter of $8$ feet is positioned in the center of a square lawn. The lawn has a side length of $20$ feet. A concrete walkway of width $2$ feet is built directly around the fountain. What is the area, in square feet, of the remaining grass region of the lawn?

400 - 36\pi

400 - 16\pi

400 - 24\pi

400 - 100\pi

Cevabı ve açıklamayı göster

Cevap: $400 - 36\pi$

Cevap

400 - 36\pi

The area of the square lawn is

20^2 = 400

square feet. The fountain has a radius of

4

feet (half of the

8

-foot diameter), and the walkway adds another

2

feet to the radius, making the total radius of the combined circular area

6

feet. The area of this circular region is

\pi \times 6^2 = 36\pi

square feet. Subtracting this from the total area of the square lawn gives the remaining grass area of

400 - 36\pi

square feet.

Adım Adım Çözüm

1

Calculate the total area of the square lawn.

Area of the lawn is

20 \times 20 = 400

square feet.

The area of a square is calculated by squaring its side length.

2

Find the combined radius of the fountain and the walkway.

The radius of the fountain is

8 \div 2 = 4

feet. Adding the

2

-foot width of the walkway gives a combined radius of

4 + 2 = 6

feet.

The radius is half of the diameter. The walkway surrounds the fountain, so its width must be added to the fountain's radius to find the outer boundary's radius.

3

Calculate the combined area of the fountain and the walkway.

Area of the combined circular region is

\pi \times 6^2 = 36\pi

square feet.

The area of a circle is given by

\pi r^2

, where

r

is the radius.

4

Subtract the combined circular area from the total area of the lawn to find the remaining grass area.

The remaining area is

400 - 36\pi

square feet.

The remaining grass area is the total square area minus the area of the inner circular region that contains the fountain and the walkway.

Anahtar Kavram

Area of composite shapes involving squares and circles, and scaling properties.

Tahmini Süre:1m 30s

Soru 17Soru →

An L-shaped region is created by removing a smaller square from the corner of a larger square. The perimeter of the L-shaped region is $48$ inches, and the area of the smaller square that was removed is $16$ square inches. What is the area, in square inches, of the L-shaped region?

Cevabı ve açıklamayı göster

Cevap: 128

Cevap

128

The area of the L-shaped region is the difference between the area of the original larger square and the area of the removed corner square. The perimeter of an L-shaped region formed by removing a corner square is identical to the perimeter of the original square, which is

48

inches. This means the side length of the larger square is

12

inches, and its area is

144

square units. Subtracting the area of the removed square (

16

square units) from the area of the larger square gives

128

square units.

Adım Adım Çözüm

1

Calculate the side length of the smaller square from its area.

The side length of the smaller square is

4

inches.

The area of a square is the square of its side length (

A = s^2

). Since the area is

16

, we have

s = \sqrt{16} = 4

.

2

Find the side length of the larger square using the perimeter of the L-shaped region.

The side length of the larger square is

12

inches.

When a corner square is removed from a larger square, the perimeter remains unchanged because the two cut-out edges going inward have the same lengths as the two outer edges that were removed. Thus, the perimeter of the L-shaped region is equal to

4S

. With a perimeter of

48

, the side length

S

is

48 \div 4 = 12

.

3

Compute the area of the L-shaped region.

The area of the L-shaped region is

128

square inches.

The area of the L-shaped region is the area of the larger square minus the area of the removed smaller square:

12^2 - 16 = 144 - 16 = 128

.

Anahtar Kavram

The area of a composite shape can be calculated by subtracting the area of a removed sub-region from the area of the outer boundary. The perimeter of a rectangle or square remains unchanged when a corner square is removed.

Soru 18Soru →

A logo design consists of a region bounded by an isosceles trapezoid and a semicircle. The semicircle is attached to the shorter parallel side of the trapezoid such that the diameter of the semicircle is equal to the length of that side, and the semicircle lies entirely outside the trapezoid. The trapezoid has a height of $8$ centimeters, a longer parallel side of $18$ centimeters, and a shorter parallel side of $10$ centimeters. What is the total area, in square centimeters, of the logo?

224 + 12.5\pi

112 + 50\pi

112 + 12.5\pi

112 + 25\pi

Cevabı ve açıklamayı göster

Cevap: $112 + 12.5\pi$

Cevap

The total area of the logo is

112 + 12.5\pi

square centimeters.

The total area of the logo is the sum of the area of the trapezoid and the area of the semicircle. The area of the trapezoid is calculated using the formula

A = \frac{b_1 + b_2}{2}h

, which gives

\frac{18 + 10}{2} \times 8 = 112

square centimeters. The area of the semicircle is half the area of a full circle with a diameter of

10

centimeters (radius of

5

centimeters), which is

\frac{1}{2}\pi r^2 = \frac{1}{2}\pi (5^2) = 12.5\pi

square centimeters. Adding these two areas together gives

112 + 12.5\pi

square centimeters.

Adım Adım Çözüm

1

Calculate the area of the trapezoid section of the logo.

The area of the trapezoid is

112

square centimeters.

The area of a trapezoid is given by the formula

A = \frac{b_1 + b_2}{2}h

, where

b_1 = 18

,

b_2 = 10

, and

h = 8

.

2

Calculate the area of the semicircle section of the logo.

The area of the semicircle is

12.5\pi

square centimeters.

The diameter of the semicircle is the shorter base of the trapezoid (

10

centimeters), so the radius is

r = 5

centimeters. The area of a semicircle is half the area of a full circle:

A = \frac{1}{2}\pi r^2 = \frac{1}{2}\pi (5)^2 = 12.5\pi

.

3

Sum the areas of the trapezoid and the semicircle to find the total area.

The total area is

112 + 12.5\pi

square centimeters.

The total area of a composite figure is the sum of the areas of its non-overlapping component parts.

Anahtar Kavram

Calculating the total area of a composite figure by decomposing it into standard shapes (a trapezoid and a semicircle) and summing their individual areas.

Tahmini Süre:1m 30s

Soru 19Soru →

A designer has a rectangular piece of fabric that measures $12$ inches by $18$ inches. The designer cuts out two identical right triangular pieces from the corners along one of the $12$ -inch sides. Each right triangular piece has a leg of length $4$ inches along the $12$ -inch side and a leg of length $x$ inches along the $18$ -inch side. If the area of the remaining piece of fabric is $180$ square inches, what is the value of $x$ ?

Cevabı ve açıklamayı göster

Cevap: 9

Cevap

9

The correct answer is 9. The original area of the rectangular fabric is

12 \times 18 = 216

square inches. Two identical right triangles with leg lengths of

4

inches and

x

inches are cut out. The area of each triangle is

\frac{1}{2} \times 4 \times x = 2x

square inches. The total area of the two triangles is

2 \times 2x = 4x

square inches. Subtracting this from the original area gives the remaining area:

216 - 4x = 180

. Solving for

x

gives

4x = 36

, which simplifies to

x = 9

.

Adım Adım Çözüm

1

Calculate the area of the original rectangular piece of fabric.

216 square inches

To find the initial area before any modifications are made, using the formula

\text{Area} = \text{length} \times \text{width}

.

2

Find the total area of the two cut-out right triangles in terms of

x

.

4x

square inches

Each right triangle has legs of

4

and

x

, so its area is

\frac{1}{2}(4)(x) = 2x

. The total area of two such identical triangles is

2(2x) = 4x

.

3

Set up an equation using the remaining area of the fabric.

216 - 4x = 180

The remaining area of

180

square inches is equal to the original area of

216

square inches minus the total area of the two cut-out triangles, which is

4x

.

4

Solve the equation for

x

.

x = 9

Isolating the variable

x

by subtracting

216

from both sides and then dividing by

-4

yields

x = 9

.

Anahtar Kavram

Area of composite shapes (rectangles and triangles)

Soru 20Soru →

A property line is in the shape of a trapezoid with parallel sides of length $20$ meters and $30$ meters, and a height of $12$ meters. A second property is geometrically similar to the first, where each linear dimension of the second property is $1.5$ times the corresponding dimension of the first property. What is the area, in square meters, of the second property?

450

675

900

1,350

Cevabı ve açıklamayı göster

Cevap: 675

Cevap

675

The area of the first property is calculated using the trapezoid area formula:

A = \frac{1}{2}(b_1 + b_2)h

. Substituting the given values gives

A = \frac{1}{2}(20 + 30)(12) = 300

square meters. Since the second property is similar to the first with a linear scale factor of

1.5

, its area is scaled by

1.5^2 = 2.25

. Therefore, the area of the second property is

300 \times 2.25 = 675

square meters.

Adım Adım Çözüm

1

Calculate the area of the first trapezoidal property.

300

square meters

Using the trapezoid area formula

A = \frac{1}{2}(b_1 + b_2)h

, we find

A = \frac{1}{2}(20 + 30)(12) = 300

.

2

Determine the area scale factor for the similar property.

2.25

Since the second property is similar to the first with a linear scale factor of

1.5

, its area scales by the square of this factor:

(1.5)^2 = 2.25

.

3

Calculate the area of the second property.

675

square meters

Multiplying the original area by the area scale factor yields

300 \times 2.25 = 675

.

Anahtar Kavram

Area of similar two-dimensional shapes scales by the square of the linear scale factor.

Area of Two-Dimensional Shapes

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Alternatif Yöntem

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Alternatif Yöntem

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