Geometry Alıştırma Soruları — Armed Services Vocational Aptitude Battery (ASVAB)

A circular swimming pool has a circumference of $20\pi$ feet. What is the diameter of the pool in feet?

Cevabı ve açıklamayı göster

Cevap: 20

Cevap

20

The formula for the circumference of a circle is

C = \pi d

. By substituting the given circumference of

20\pi

into the formula, we get

20\pi = \pi d

. Dividing both sides by

\pi

yields a diameter of

20

feet.

Adım Adım Çözüm

1

Identify the formula for the circumference of a circle.

C = \pi d

, where

C

is circumference and

d

is diameter.

This formula directly relates the given circumference to the unknown diameter.

2

Substitute the given value into the formula.

20\pi = \pi d

The problem states the circumference is

20\pi

feet.

3

Solve for the diameter (

d

).

d = 20

Dividing both sides of the equation by

\pi

isolates the value of

d

.

Anahtar Kavram

Circumference of a circle

Tahmini Süre:45s

Soru 2Soru →

Determine the missing side length based on the geometric properties of a right triangle.

Aşağıdaki boşlukları doldurun

In a right triangle, the two shorter sides (legs) measure

9

cm and

12

cm. According to the Pythagorean theorem, the length of the hypotenuse is cm.

Cevabı ve açıklamayı göster

Cevap

15

By applying the Pythagorean theorem (

a^2 + b^2 = c^2

) to a right triangle with legs of

9

and

12

, we square both numbers to get

81

and

144

. Their sum is

225

. The hypotenuse is the square root of

225

, which is

15

.

Adım Adım Çözüm

1

Identify the known sides of the right triangle.

The legs (let's call them

a

and

b

) measure

9

cm and

12

cm.

The Pythagorean theorem requires knowing the lengths of two sides to find the third.

2

Set up the Pythagorean theorem equation.

a^2 + b^2 = c^2

, which becomes

9^2 + 12^2 = c^2

.

This formula relates the squares of the legs to the square of the hypotenuse (

c

).

3

Square the lengths of the two legs and add them together.

81 + 144 = 225

This calculation determines the value of

c^2

.

4

Take the square root of the sum to find the hypotenuse.

\sqrt{225} = 15

Since

c^2 = 225

, taking the square root gives the actual length of the hypotenuse (

c

).

Anahtar Kavram

Pythagorean Theorem

Soru 3Soru →

A rectangular swimming pool measures $40$ feet in length and $30$ feet in width. A concrete deck of uniform width is built around the entire outer edge of the pool. If the area of the concrete deck alone is $296$ square feet, what is the width of the deck, in feet?

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Cevap: 2

Cevap

2

The correct width is found by setting up an equation for the total area. The deck adds its width to both sides of the pool's length and width, making the overall dimensions

40 + 2w

and

30 + 2w

. The area of the deck alone is the total area minus the pool's area. Solving the algebraic equation

(40 + 2w)(30 + 2w) - 1200 = 296

yields a positive root of

w = 2

.

Adım Adım Çözüm

1

Calculate the area of the pool.

40 \times 30 = 1200

square feet.

You need the baseline area of the inner rectangle before the deck is added.

2

Write algebraic expressions for the outer dimensions of the pool and deck combined.

Length

= 40 + 2w

, Width

= 30 + 2w

.

The deck has a uniform width

w

, which adds to both the top/bottom and left/right sides of the pool.

3

Formulate an equation for the area of the deck.

(40 + 2w)(30 + 2w) - 1200 = 296

.

The area of the deck alone is equal to the total area (pool plus deck) minus the area of the pool.

4

Expand the binomials and simplify into a standard quadratic equation.

1200 + 140w + 4w^2 - 1200 = 296 \implies 4w^2 + 140w - 296 = 0

.

Standard quadratic form (

ax^2 + bx + c = 0

) is required to solve for the unknown variable

w

.

5

Solve the quadratic equation for

w

.

Divide by

4

to get

w^2 + 35w - 74 = 0

. Factoring yields

(w + 37)(w - 2) = 0

, so

w = 2

.

Finding the positive root gives the actual physical width of the concrete deck.

Anahtar Kavram

Calculating the area of composite rectangular figures and solving quadratic equations derived from expanding geometric boundaries.

Soru 4Soru →

An automated lawn sprinkler rotates through an angle of $120^\circ$ and sprays water outward to a maximum distance of $18$ feet. Which of the following represents the total area of the lawn covered by this sprinkler? (Leave your answer in terms of $\pi$ .)

108\pi \text{ sq ft}

27\pi \text{ sq ft}

12\pi \text{ sq ft}

6\pi \text{ sq ft}

Cevabı ve açıklamayı göster

Cevap: $108\pi \text{ sq ft}$

Cevap

108\pi \text{ sq ft}

The sprinkler covers a sector of a circle. The radius of this circle is the maximum spray distance, which is

18

feet. The full area of a circle with this radius would be

\pi(18)^2 = 324\pi

square feet. Since the sprinkler rotates

120^\circ

, which is

\frac{120}{360}

or

\frac{1}{3}

of a full circle, the covered area is

\frac{1}{3}

of

324\pi

, resulting in

108\pi

square feet.

Adım Adım Çözüm

1

Identify the geometric shape and the relevant area formula.

The area covered by the sprinkler is a sector of a circle. The formula for the area of a sector is

Area = \frac{\theta}{360^\circ} \times \pi r^2

.

The sprinkler covers a fractional portion of a full circle defined by its rotation angle.

2

Determine the fraction of the circle covered.

\frac{120^\circ}{360^\circ} = \frac{1}{3}

The sprinkler rotates

120^\circ

out of a total possible

360^\circ

for a complete circle.

3

Substitute the radius into the formula and solve.

\frac{1}{3} \times \pi \times (18)^2 = \frac{1}{3} \times \pi \times 324 = 108\pi

The maximum spray distance (

18

feet) is the radius. Squaring

18

gives

324

, and finding one-third of this yields the sector's area.

Anahtar Kavram

Area of a Circular Sector

Soru 5Soru →

A stained glass artist cuts a piece of glass into the shape of a right triangle. The two perpendicular legs of the triangle measure $30$ centimeters and $40$ centimeters. What is the total perimeter of this glass triangle in centimeters?

Cevabı ve açıklamayı göster

Cevap: 120

Cevap

120

The correct answer is found by first determining the length of the hypotenuse using the Pythagorean theorem. Since the legs are

30

and

40

, the hypotenuse is

\sqrt{30^2 + 40^2} = \sqrt{2500} = 50

. The perimeter is then calculated by adding the lengths of all three sides:

30 + 40 + 50 = 120

.

Adım Adım Çözüm

1

Use the Pythagorean theorem to find the length of the missing hypotenuse.

c = \sqrt{30^2 + 40^2} = \sqrt{900 + 1600} = \sqrt{2500} = 50

centimeters.

The perimeter requires all three side lengths, but only the two legs are provided.

2

Calculate the total perimeter of the right triangle.

Perimeter =

30 + 40 + 50 = 120

centimeters.

The perimeter is the total distance around the outside of the shape, found by summing all side lengths.

Anahtar Kavram

Applying the Pythagorean theorem to find a missing side length, then using all three side lengths to calculate the perimeter of a right triangle.

Soru 6Soru →

Determine the required dimensions for the construction project.

Aşağıdaki boşlukları doldurun

A construction crew is building a wooden ramp with a right triangular cross-section. The horizontal base of the ramp measures

24\text{ feet}

and it rises to a vertical height of

7\text{ feet}

. According to the Pythagorean Theorem, the length of the sloped surface is feet, and the total perimeter of the ramp's triangular profile is feet.

Cevabı ve açıklamayı göster

Cevap

The sloped surface is 25 feet, and the total perimeter is 56 feet.

The sloped surface corresponds to the hypotenuse of a right triangle with legs of

7

and

24

feet. Applying the Pythagorean Theorem,

7^2 + 24^2 = 49 + 576 = 625

. Taking the square root of

625

gives

25\text{ feet}

for the sloped surface. The total perimeter is the sum of the base, height, and sloped surface:

24 + 7 + 25 = 56\text{ feet}

.

Adım Adım Çözüm

1

Identify the known sides of the right triangle.

The legs of the right triangle are

a = 7

and

b = 24

.

The horizontal base and vertical height form the two legs that intersect at a right angle.

2

Apply the Pythagorean Theorem to find the length of the hypotenuse.

c^2 = 7^2 + 24^2 = 49 + 576 = 625

. Therefore,

c = \sqrt{625} = 25

.

The sloped surface represents the hypotenuse (

c

) of the right triangle.

3

Calculate the perimeter of the triangle.

7 + 24 + 25 = 56

.

The perimeter is found by summing the lengths of all three sides of the triangular cross-section.

Anahtar Kavram

Using the Pythagorean Theorem to find the hypotenuse of a right triangle, and applying it to calculate the perimeter.

Soru 7Soru →

What is the measure of one interior angle of a regular octagon?

45^\circ

135^\circ

180^\circ

225^\circ

Cevabı ve açıklamayı göster

Cevap: $135^\circ$

Cevap

The measure of one interior angle of a regular octagon is

135^\circ

.

To find a single interior angle of a regular polygon, use the formula

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

. Substituting

n = 8

for an octagon gives

\frac{6 \times 180^\circ}{8} = \frac{1080^\circ}{8} = 135^\circ

.

Adım Adım Çözüm

1

Identify the number of sides (

n

) for an octagon.

n = 8

An octagon is a polygon with 8 sides.

2

Calculate the total sum of the interior angles using the formula

(n - 2) \times 180^\circ

.

(8 - 2) \times 180^\circ = 6 \times 180^\circ = 1080^\circ

This formula determines the combined measure of all interior angles for any polygon.

3

Divide the total sum by the number of sides to find the measure of a single angle.

1080^\circ \div 8 = 135^\circ

In a regular polygon, all interior angles are equal, so dividing the total sum by the number of angles gives the measure of one.

Anahtar Kavram

Interior Angles of Regular Polygons

Soru 8Soru →

The interior angles of a quadrilateral are represented by the algebraic expressions $2x^\circ$ , $(3x - 15)^\circ$ , $(x + 25)^\circ$ , and $(2x + 30)^\circ$ . What is the measure of the largest angle in this quadrilateral?

135^\circ

40^\circ

110^\circ

65^\circ

Cevabı ve açıklamayı göster

Cevap: $110^\circ$

Cevap

The largest angle measures

110^\circ

.

The interior angles of a quadrilateral must sum to

360^\circ

. By adding the four expressions together, we get

8x + 40 = 360

. Solving for

x

yields

x = 40

. To find the largest angle, we substitute

40

back into the original expressions:

2(40) = 80^\circ

,

3(40) - 15 = 105^\circ

,

40 + 25 = 65^\circ

, and

2(40) + 30 = 110^\circ

. Comparing these four values,

110^\circ

is the largest.

Adım Adım Çözüm

1

Set up an equation based on the sum of interior angles.

2x + (3x - 15) + (x + 25) + (2x + 30) = 360

The sum of the interior angles of any quadrilateral is always

360^\circ

.

2

Combine the like terms on the left side of the equation.

8x + 40 = 360

Grouping the

x

terms (

2x + 3x + x + 2x = 8x

) and constant terms (

-15 + 25 + 30 = 40

) simplifies the algebraic expression.

3

Solve for the variable

x

.

8x = 320

, which means

x = 40

.

Subtracting 40 from both sides and then dividing by 8 isolates

x

.

4

Substitute

x = 40

into each of the original expressions to determine the measure of each angle.

The angles are

2(40) = 80^\circ

;

3(40) - 15 = 105^\circ

;

40 + 25 = 65^\circ

; and

2(40) + 30 = 110^\circ

.

The value of

x

alone does not answer the question; the specific angle measures must be evaluated.

5

Identify the largest value among the calculated angles.

The largest value is

110^\circ

.

The question specifically asks for the maximum angle measure in the quadrilateral.

Anahtar Kavram

The sum of the interior angles of a quadrilateral is always 360 degrees. Algebraic substitution is required to convert variable expressions back into geometric measurements.

Tahmini Süre:1m 30s

Soru 9Soru →

Match each polygon or quadrilateral with its defining geometric property.

Soldaki öğeye tıklayın, sonra eşleşen sağdaki öğeye tıklayın

Cevabı ve açıklamayı göster

Cevap

Parallelogram matches with 'two pairs of parallel opposite sides', Trapezoid matches with 'exactly one pair of parallel sides', Rhombus matches with 'four sides of equal length', and Hexagon matches with 'exactly six sides'.

Each shape is matched to its fundamental geometric definition based on the number of sides, side lengths, and parallel properties. Knowing these core definitions is essential for classifying polygons and quadrilaterals correctly.

Adım Adım Çözüm

1

Analyze the properties of a parallelogram.

It has two pairs of parallel opposite sides.

This is the fundamental geometric definition of a parallelogram.

2

Analyze the properties of a trapezoid.

It is defined by having exactly one pair of parallel sides.

This distinguishes a trapezoid from other quadrilaterals like parallelograms.

3

Analyze the properties of a rhombus.

It is a quadrilateral where all four sides are of equal length.

While it also has parallel sides, the equal side lengths are its distinguishing feature.

4

Identify the number of sides for a hexagon.

A hexagon is a polygon with exactly six sides.

The prefix 'hexa-' denotes six in geometry.

Anahtar Kavram

Basic Properties of Quadrilaterals and Polygons

Soru 10Soru →

A decorative floor tile is manufactured in the shape of a regular pentagon. What is the degree measure of each interior angle of the tile?

72^\circ

108^\circ

180^\circ

540^\circ

Cevabı ve açıklamayı göster

Cevap: $108^\circ$

Cevap

108^\circ

The interior angles of an

n

-sided polygon sum to

(n - 2) \times 180^\circ

. For a 5-sided pentagon, the sum is

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

. Since the tile is a regular pentagon, all five angles are identical. Dividing the total

540^\circ

by 5 yields exactly

108^\circ

per corner.

Adım Adım Çözüm

1

Identify the number of sides in a regular pentagon.

n = 5

A pentagon is defined as a geometric polygon with exactly 5 sides.

2

Calculate the sum of all interior angles using the polygon formula

(n - 2) \times 180^\circ

.

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

This formula determines the total degree measure for the interior angles of any convex polygon.

3

Divide the total sum by the number of sides to find the measure of a single interior angle.

540^\circ \div 5 = 108^\circ

Because it is a regular pentagon, all interior angles are equal in measure.

Anahtar Kavram

Interior Angles of Regular Polygons

Soru 11Soru →

Match each quadrilateral with its precise geometric defining properties.

Soldaki öğeye tıklayın, sonra eşleşen sağdaki öğeye tıklayın

Cevabı ve açıklamayı göster

Cevap

Rhombus matches the parallelogram with perpendicular diagonals that bisect angles. Rectangle matches the parallelogram with four right angles. Kite matches the quadrilateral with consecutive congruent sides. Isosceles Trapezoid matches the quadrilateral with exactly one pair of parallel sides.

Each quadrilateral has unique defining traits involving their sides, angles, and diagonals. A rhombus requires perpendicular diagonals and parallel opposite sides. A rectangle requires right angles and congruent diagonals. A kite requires adjacent equal sides and perpendicular diagonals. An isosceles trapezoid requires exactly one pair of parallel sides and congruent diagonals.

Adım Adım Çözüm

1

Analyze the unique properties of a Rhombus.

It is a parallelogram with all equal sides, which leads to perpendicular diagonals that bisect the opposite angles.

To distinguish it from other standard parallelograms like rectangles.

2

Analyze the unique properties of a Rectangle.

It has four right angles and, unlike a standard parallelogram, has diagonals that are identical in length.

To correctly identify the item describing congruent diagonals in a parallelogram.

3

Analyze the unique properties of a Kite.

It has adjacent sides that are equal (consecutive congruent sides), which forces the diagonals to meet at 90 degrees.

To match it with the correct non-parallelogram definition.

4

Analyze the unique properties of an Isosceles Trapezoid.

It has exactly one pair of parallel sides (bases) and the symmetry from equal non-parallel sides creates equal diagonals.

To differentiate it from general trapezoids and generic parallelograms.

Anahtar Kavram

Quadrilaterals and Polygons

Soru 12Soru →

A land surveyor is measuring a four-sided plot of land. They record three of the interior angles as $78^\circ$ , $104^\circ$ , and $83^\circ$ . What is the measure of the fourth interior angle, in degrees?

Cevabı ve açıklamayı göster

Cevap: 95

Cevap

The fourth interior angle measures

95^\circ

.

The correct answer is found by subtracting the sum of the three known angles (

265^\circ

) from the total interior angle sum of a quadrilateral (

360^\circ

), which yields

95^\circ

.

Adım Adım Çözüm

1

Identify the total sum of interior angles for a quadrilateral.

360^\circ

A quadrilateral has 4 sides, and the sum of interior angles is calculated as

(n - 2) \times 180^\circ

. For

n = 4

, this is

2 \times 180^\circ = 360^\circ

.

2

Calculate the sum of the three given angles.

78^\circ + 104^\circ + 83^\circ = 265^\circ

To find the missing part, we must first determine the combined total of the known angles.

3

Subtract the sum of the given angles from the total quadrilateral angle sum.

360^\circ - 265^\circ = 95^\circ

The remaining amount represents the exact measure of the fourth interior angle.

Anahtar Kavram

The sum of the interior angles of any convex quadrilateral is always exactly

360^\circ

.

Soru 13Soru →

A rectangular room has a length of $12$ feet and a width of $10$ feet. What is the perimeter of the room?

22

feet

34

feet

44

feet

120

feet

Cevabı ve açıklamayı göster

Cevap: $44$ feet

Cevap

44

feet

The perimeter is the total distance around the boundary of the rectangle. Since a rectangle has two equal lengths and two equal widths, the perimeter is

2 \times 12 + 2 \times 10 = 24 + 20 = 44

feet.

Adım Adım Çözüm

1

Identify the formula for the perimeter of a rectangle.

P = 2 \times (\text{Length} + \text{Width})

or

P = 2L + 2W

The perimeter is the total distance around the outside of the shape, which includes two lengths and two widths.

2

Substitute the given dimensions into the formula.

P = 2 \times (12 + 10)

The length is given as

12

feet and the width is

10

feet.

3

Calculate the final perimeter.

P = 2 \times 22 = 44

Performing the addition inside the parentheses first, then multiplying by

2

yields the total distance.

Anahtar Kavram

The perimeter of a rectangle is the total distance around its edges, calculated as twice the sum of its length and width.

Tahmini Süre:45s

Soru 14Soru →

In a parallelogram, the measure of one interior angle is $30^\circ$ less than twice the measure of the consecutive interior angle. What is the measure of the larger angle?

110^\circ

50^\circ

130^\circ

230^\circ

Cevabı ve açıklamayı göster

Cevap: $110^\circ$

Cevap

The measure of the larger angle is

110^\circ

.

The correct answer correctly identifies that consecutive angles in a parallelogram are supplementary. Setting up the equation

x + (2x - 30) = 180^\circ

and solving for

x

yields

70^\circ

. Substituting this back into the expression for the larger angle gives

2(70) - 30 = 110^\circ

.

Adım Adım Çözüm

1

Identify the geometric property relating the angles.

Consecutive interior angles of a parallelogram are supplementary, meaning they add up to

180^\circ

.

This is a fundamental property of parallelograms formed by parallel lines cut by a transversal.

2

Translate the word problem into an algebraic equation.

Let the smaller angle be

x

. The larger angle is

2x - 30

. Therefore,

x + (2x - 30) = 180

.

This captures both the relationship described ('

30^\circ

less than twice') and the supplementary geometric property.

3

Solve the algebraic equation for

x

.

Combine terms:

3x - 30 = 180

. Add

30

to both sides:

3x = 210

. Divide by

3

:

x = 70^\circ

.

Isolating the variable determines the exact measure of the smaller angle.

4

Calculate the measure of the larger angle.

2(70) - 30 = 140 - 30 = 110^\circ

.

The question specifically asks for the measure of the larger angle, not the smaller one.

Anahtar Kavram

Properties of Parallelograms and Algebraic Translation

Alternatif Yöntem

You can solve this by testing the answer choices. If the larger angle is

110^\circ

, the smaller consecutive angle must be

180^\circ - 110^\circ = 70^\circ

. Check if

110^\circ

is

30^\circ

less than twice

70^\circ

:

2(70) - 30 = 140 - 30 = 110^\circ

. The values perfectly match the problem statement.

Tahmini Süre:1m 15s

Soru 15Soru →

A triangular pennant has a base of $8$ inches and a height of $15$ inches. What is the area of the pennant, in square inches?

Cevabı ve açıklamayı göster

Cevap: 60

Cevap

60

The correct area is found by multiplying the base by the height, and then dividing by two according to the standard triangle area formula.

Adım Adım Çözüm

1

Identify the formula for the area of a triangle.

A = \frac{1}{2}bh

The object is described as a triangular pennant.

2

Substitute the given base and height into the formula.

A = \frac{1}{2}(8)(15)

The base is given as

8

inches and the height is

15

inches.

3

Calculate the final product.

60

Multiplying the terms:

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

, and

4 \times 15 = 60

.

Anahtar Kavram

Area of a Triangle

Tahmini Süre:45s

Soru 16Soru →

The sum of the interior angles of a convex polygon is exactly three times the sum of its exterior angles. How many sides does this polygon have?

Cevabı ve açıklamayı göster

Cevap: 8

Cevap

8

Applying the core polygon properties, the interior angle sum is

180^\circ(n - 2)

and the exterior angle sum is

360^\circ

. Setting

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

and solving for

n

yields 8.

Adım Adım Çözüm

1

Identify the interior angle sum formula.

Sum =

180^\circ(n - 2)

This formula provides the total sum of all interior angles for an

n

-sided polygon.

2

Identify the exterior angle sum property.

Sum =

360^\circ

A fundamental property of all convex polygons is that the sum of their exterior angles is exactly

360^\circ

.

3

Set up the algebraic equation.

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

The question states the interior sum is exactly three times the exterior sum.

4

Solve the equation for

n

.

n = 8

Dividing

1080^\circ

by

180^\circ

yields 6, and adding 2 gives a final side count of 8.

Anahtar Kavram

Polygon Angle Sum Theorems

Soru 17Soru →

A circular target has a diameter of $16$ inches. Which of the following represents the area of the target, in square inches?

64\pi

256\pi

16\pi

32\pi

Cevabı ve açıklamayı göster

Cevap: $64\pi$

Cevap

64\pi

The correct answer accurately divides the diameter of

16

inches by

2

to find the radius of

8

inches. It then squares the radius (

8^2 = 64

) and multiplies by

\pi

to find the area of

64\pi

square inches.

Adım Adım Çözüm

1

Find the radius of the circle.

r = 16 / 2 = 8

inches

The area formula requires the radius, which is exactly half of the diameter.

2

Apply the area formula for a circle (

A = \pi r^2

).

A = \pi(8)^2

Substituting the calculated radius into the standard area formula.

3

Simplify the expression.

A = 64\pi

square inches

Squaring the radius (

8 \times 8 = 64

) provides the final area in terms of

\pi

.

Anahtar Kavram

The area of a circle is found using the formula

A = \pi r^2

. When given the diameter, it must first be divided by 2 to find the radius.

Soru 18Soru →

Two parallel lines, $L_1$ and $L_2$ , are intersected by three distinct transversals. The first transversal creates a pair of alternate interior angles that measure $(4x - 3y)^\circ$ and $(2x + y)^\circ$ . The second transversal creates a pair of consecutive (same-side) interior angles that measure $(3x + y)^\circ$ and $(5y + 36)^\circ$ . The third transversal creates a pair of corresponding angles. One of these corresponding angles measures $(x + y + z)^\circ$ , and the angle forming a linear pair with it measures $108^\circ$ . What is the value of $z$ ?

Cevabı ve açıklamayı göster

Cevap: 36

Cevap

36

By applying geometric theorems for transversals (alternate interior angles are equal, same-side interior angles sum to 180), a system of equations reveals x = 24 and y = 12. Using the linear pair property, the final angle expression (x + y + z) equals 72, which isolates z to exactly 36.

Adım Adım Çözüm

1

Set up an equation using the alternate interior angles.

4x - 3y = 2x + y \Rightarrow 2x = 4y \Rightarrow x = 2y

Alternate interior angles formed by a transversal intersecting parallel lines are equal.

2

Set up an equation using the consecutive (same-side) interior angles.

(3x + y) + (5y + 36) = 180 \Rightarrow 3x + 6y + 36 = 180 \Rightarrow 3x + 6y = 144

Consecutive interior angles formed by a transversal intersecting parallel lines are supplementary (sum to 180 degrees).

3

Solve the system of equations by substituting

x = 2y

into the second equation.

3(2y) + 6y = 144 \Rightarrow 6y + 6y = 144 \Rightarrow 12y = 144 \Rightarrow y = 12

. Since

x = 2y

,

x = 24

.

Substitution efficiently solves a two-variable linear system to find the individual angle components.

4

Use the linear pair to determine the measure of the corresponding angle, then solve for

z

.

(x + y + z) = 180 - 108 = 72

. Substituting known values:

24 + 12 + z = 72 \Rightarrow 36 + z = 72 \Rightarrow z = 36

.

Angles forming a linear pair are supplementary. Finding the target angle allows substitution of

x

and

y

to isolate

z

.

Anahtar Kavram

Complex systems of transversal angle relationships (alternate interior, consecutive interior, linear pairs)

Tahmini Süre:3m 0s

Soru 19Soru →

A convex hexagon has five interior angles that measure $120^\circ$ , $135^\circ$ , $115^\circ$ , $140^\circ$ , and $110^\circ$ . If the measure of the sixth interior angle is represented by the algebraic expression $(4x + 16)^\circ$ , what is the value of $x$ ?

5

21

29

111

Cevabı ve açıklamayı göster

Cevap: $21$

Cevap

The correct answer is

21

.

The sum of the interior angles of a hexagon (

6

sides) is

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

. The five given angles add up to

620^\circ

. This means the sixth angle must be

720^\circ - 620^\circ = 100^\circ

. Setting the algebraic expression

4x + 16

equal to

100

and solving for

x

by subtracting

16

and dividing by

4

gives

x = 21

.

Adım Adım Çözüm

1

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

The sum is

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

.

The formula for the sum of interior angles is

(n - 2) \times 180^\circ

, where

n

is the number of sides. A hexagon has

6

sides.

2

Find the sum of the five given angles.

120^\circ + 135^\circ + 115^\circ + 140^\circ + 110^\circ = 620^\circ

.

Adding the known angles allows us to find the combined measure of the known portion of the polygon.

3

Calculate the measure of the sixth angle.

720^\circ - 620^\circ = 100^\circ

.

Subtracting the sum of the five known angles from the total sum gives the measure of the missing angle.

4

Set up and solve the equation for

x

.

4x + 16 = 100 \implies 4x = 84 \implies x = 21

.

The sixth angle is represented by the expression

(4x + 16)^\circ

, so setting it equal to

100

allows us to isolate and solve for

x

.

Anahtar Kavram

Polygon Interior Angles and Linear Equations

Soru 20Soru →

The perimeter of a square courtyard is $36$ meters. What is the area of the courtyard, in square meters?

Cevabı ve açıklamayı göster

Cevap: 81

Cevap

81

A square has four sides of equal length. Given a perimeter of

36

meters, each side measures

36 \div 4 = 9

meters. The area of a square is its side length squared, which is

9^2 = 81

square meters.

Adım Adım Çözüm

1

Find the length of one side of the square.

36 \div 4 = 9

meters.

The perimeter of a square is the sum of its four equal sides, so dividing the total perimeter by 4 gives the length of a single side.

2

Calculate the area of the square.

9 \times 9 = 81

square meters.

The area of a square is calculated by squaring the length of one of its sides (

A = s^2

).

Anahtar Kavram

Perimeter and Area of 2D Figures

Geometry

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