Properties of Quadrilaterals Practice Questions — ACT — Page 2

Question 21Question →

In any parallelogram $ABCD$ , if consecutive angles $\angle A$ and $\angle B$ are supplementary, then quadrilateral $ABCD$ must be a rectangle.

True

False

Show answer & explanation

Answer: False

Answer

False. Consecutive interior angles are supplementary in all parallelograms, not only in rectangles.

The statement is false because consecutive interior angles are supplementary in every parallelogram due to parallel opposite sides. This property does not imply that the angles are right angles (

90^\circ

); for instance, a parallelogram with angles measuring

60^\circ

and

120^\circ

has supplementary consecutive angles but is clearly not a rectangle.

Step-by-Step Solution

1

Recall the consecutive angle property for any general parallelogram.

In any parallelogram

ABCD

, opposite sides are parallel (

AD \parallel BC

). When parallel lines are intersected by a transversal

AB

, consecutive interior angles are supplementary:

\angle A + \angle B = 180^\circ

.

Consecutive interior angles formed by parallel lines and a transversal always sum to

180^\circ

.

2

Compare this general property with the specific condition that defines a rectangle.

A parallelogram is a rectangle if and only if all four interior angles are right angles (

90^\circ

), which requires consecutive angles to be congruent (equal in measure), not merely supplementary.

Two supplementary angles can have measures such as

60^\circ

and

120^\circ

, which form an oblique parallelogram rather than a rectangle.

3

Determine the truth value of the statement.

Because the condition of having supplementary consecutive angles is satisfied by every parallelogram and does not guarantee

90^\circ

angles, the statement is false.

A property shared by all members of a general class (parallelograms) cannot be used as a sufficient condition to classify a figure into a restrictive subclass (rectangles).

Key Concept

Properties of Quadrilaterals: Parallelogram vs. Rectangle Angle Rules

Estimated Time:1m 0s

Question 22Question →

In rhombus $ABCD$ , the measure of interior angle $\angle DAB$ is $120^\circ$ , and the length of diagonal $AC$ is $12$ inches. What is the perimeter, in inches, of rhombus $ABCD$ ?

24

48

48\sqrt{3}

72

96

Show answer & explanation

Answer: 48

Answer

The perimeter of rhombus

ABCD

is 48 inches.

Since consecutive angles in a rhombus are supplementary,

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

. Because all sides of a rhombus are equal in length,

AB = BC

, making

\triangle ABC

an isosceles triangle with a

60^\circ

vertex angle, which implies

\triangle ABC

is equilateral. Therefore, side length

AB = AC = 12

inches, and the perimeter is

4 \times 12 = 48

inches.

Step-by-Step Solution

1

Determine the consecutive angle measure in the rhombus.

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

Consecutive interior angles in a rhombus (which is a parallelogram) are supplementary.

2

Determine the nature of triangle ABC.

\triangle ABC is equilateral with AB = BC = AC = 12 inches.

A rhombus has four equal sides, so AB = BC. An isosceles triangle with a vertex angle of 60 degrees is an equilateral triangle.

3

Calculate the perimeter of the rhombus.

Perimeter = 4 \times 12 = 48 inches.

The perimeter of a rhombus is four times the length of one side.

Key Concept

Properties of a rhombus: all four sides are congruent, consecutive interior angles are supplementary, and a diagonal splitting a 60-degree angle forms an equilateral triangle with adjacent sides.

Question 23Question →

In rectangle $ABCD$ , the length of side $AB$ is $16$ centimeters and the length of side $BC$ is $12$ centimeters. Point $P$ lies on diagonal $AC$ such that segment $DP$ is perpendicular to $AC$ . What is the length, in centimeters, of segment $DP$ ?

Show answer & explanation

Answer: 9.6

Answer

The length of segment

DP

is

9.6

centimeters.

In rectangle

ABCD

, opposite sides are equal (

AD = BC = 12

cm and

DC = AB = 16

cm) and all interior angles are

90^\circ

. Right triangle

ADC

has legs

12

cm and

16

cm, making hypotenuse

AC = \sqrt{12^2 + 16^2} = 20

cm. Calculating the area of triangle

ADC

using the legs gives

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

cm². Using hypotenuse

AC

as the base and perpendicular line segment

DP

as the altitude, the area is

\frac{1}{2} \times 20 \times DP = 10 \times DP

. Setting

10 \times DP = 96

yields

DP = 9.6

cm.

Step-by-Step Solution

1

Determine the length of diagonal

AC

Diagonal

AC = 20

cm

Since

ABCD

is a rectangle, angle

ADC

is a right angle with legs

AD = 12

cm and

DC = 16

cm. By the Pythagorean theorem,

AC = \sqrt{12^2 + 16^2} = \sqrt{400} = 20

cm.

2

Calculate the area of right triangle

ADC

Area of triangle

ADC = 96

cm²

The area of a right triangle equals half the product of its legs:

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

cm².

3

Solve for altitude

DP

Length

DP = 9.6

cm

The area can also be expressed using hypotenuse

AC

as the base and

DP

as the altitude:

\text{Area} = \frac{1}{2} \times AC \times DP = \frac{1}{2} \times 20 \times DP = 10 \times DP

. Equating the two area expressions yields

10 \times DP = 96

, giving

DP = 9.6

cm.

Key Concept

Properties of Rectangles and Altitudes in Right Triangles

Question 24Question →

Determine whether the following statement is true or false: In any trapezoid, the line segment connecting the midpoints of the non-parallel sides (the midsegment) is parallel to the bases and has a length equal to half the difference of the lengths of the bases.

True

False

Show answer & explanation

Answer: False

Answer

The statement is False.

The statement is False because according to the Trapezoid Midsegment Theorem, the midsegment is parallel to the bases and its length is equal to half the sum (the arithmetic mean) of the base lengths,

m = \frac{b_1 + b_2}{2}

.

Step-by-Step Solution

1

Identify the geometric shape and segment described in the statement.

The shape is a trapezoid with parallel bases of lengths

b_1

and

b_2

, and the segment is the midsegment connecting the midpoints of the non-parallel legs.

Understanding the definition of a trapezoid's midsegment is necessary to evaluate its properties.

2

State the Trapezoid Midsegment Theorem.

The Trapezoid Midsegment Theorem states that the midsegment is parallel to both bases and its length

m

is equal to the average of the two base lengths:

m = \frac{b_1 + b_2}{2}

.

This theorem directly provides the exact mathematical relationship for the length of the midsegment.

3

Compare the theorem with the statement provided in the stem.

The statement claims the length is half the difference,

\frac{|b_1 - b_2|}{2}

, which contradicts the actual formula involving the sum.

Since the statement uses subtraction instead of addition, the statement is false.

Key Concept

Trapezoid Midsegment Theorem

Question 25Question →

In the standard $(x, y)$ coordinate plane, three consecutive vertices of parallelogram $ABCD$ are $A(-2, 1)$ , $B(3, 4)$ , and $C(5, -1)$ . If $(x, y)$ represents the coordinates of vertex $D$ , what is the value of $x + y$ ?

-4

-2

0

6

12

Show answer & explanation

Answer: $-4$

Answer

The sum of the coordinates

x + y

is

-4

.

The diagonals of a parallelogram bisect each other, meaning the midpoint of diagonal

AC

is identical to the midpoint of diagonal

BD

. Finding the midpoint of

AC

gives

\left(\frac{3}{2}, 0\right)

. Setting the midpoint of

BD

to this point gives

\frac{3+x}{2} = \frac{3}{2}

and

\frac{4+y}{2} = 0

, yielding

x = 0

and

y = -4

. The sum

x + y

is

0 + (-4) = -4

.

Step-by-Step Solution

1

Apply the diagonal midpoint property of parallelograms.

In parallelogram

ABCD

, the diagonals

AC

and

BD

bisect each other at their common midpoint

M

.

Diagonals of any parallelogram share a common midpoint.

2

Calculate the midpoint of diagonal

AC

.

M = \left(\frac{-2 + 5}{2}, \frac{1 + (-1)}{2}\right) = \left(\frac{3}{2}, 0\right)

.

The midpoint formula is

\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)

.

3

Set the midpoint of

BD

equal to

M

and solve for

x

and

y

.

\frac{3 + x}{2} = \frac{3}{2} \implies x = 0

and

\frac{4 + y}{2} = 0 \implies y = -4

. So vertex

D

is

(0, -4)

.

Equating coordinates of the common midpoint.

4

Find the sum

x + y

.

x + y = 0 + (-4) = -4

.

Adding the

x

- and

y

-coordinates of vertex

D

.

Key Concept

Properties of Parallelogram Diagonals in Coordinate Geometry

Question 26Question →

In the standard $(x, y)$ coordinate plane, trapezoid $ABCD$ has vertices at $A(0, 0)$ , $B(4, 8)$ , $C(12, 8)$ , and $D(16, 0)$ . Point $P$ is the midpoint of diagonal $AC$ , and point $Q$ is the midpoint of diagonal $BD$ . What is the distance, in coordinate units, between point $P$ and point $Q$ ?

2

4

6

8

12

Show answer & explanation

Answer: $4$

Answer

The distance between point

P

and point

Q

is

4

units.

The correct answer is

4

. Using the midpoint formula, the midpoint of diagonal

AC

is

P(6, 4)

and the midpoint of diagonal

BD

is

Q(10, 4)

. Because both points lie on the horizontal line

y = 4

, the distance between them is

|10 - 6| = 4

units.

Step-by-Step Solution

1

Calculate the coordinates of midpoint

P

of diagonal

AC

P = \left(\frac{0 + 12}{2}, \frac{0 + 8}{2}\right) = (6, 4)

The midpoint formula is

\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)

.

2

Calculate the coordinates of midpoint

Q

of diagonal

BD

Q = \left(\frac{4 + 16}{2}, \frac{8 + 0}{2}\right) = (10, 4)

Applying the midpoint formula to vertices

B(4, 8)

and

D(16, 0)

.

3

Calculate the horizontal distance between

P(6, 4)

and

Q(10, 4)

\text{Distance } PQ = \sqrt{(10 - 6)^2 + (4 - 4)^2} = 10 - 6 = 4

Since both midpoints share the same

y

-coordinate (

y = 4

), the distance is simply the absolute difference between their

x

-coordinates.

Key Concept

Midpoints of Diagonals in a Trapezoid

Alternative Method

For any trapezoid with parallel bases of lengths

b_1

and

b_2

(where

b_1 > b_2

), the length of the segment connecting the midpoints of the diagonals is given by the formula

\frac{b_1 - b_2}{2}

. Here

b_1 = 16 - 0 = 16

and

b_2 = 12 - 4 = 8

, so the length is

\frac{16 - 8}{2} = 4

.

Estimated Time:1m 15s

Question 27Question →

In kite $ABCD$ , diagonals $AC$ and $BD$ intersect perpendicularly at point $P$ . If $AP = 9$ centimeters, $PC = 16$ centimeters, and $BP = PD = 12$ centimeters, what is the perimeter, in centimeters, of kite $ABCD$ ?

Show answer & explanation

Answer: 70

Answer

The perimeter of kite

ABCD

is 70 centimeters.

The diagonals of a kite intersect at right angles (

90^\circ

). Applying the Pythagorean theorem to right triangle

APB

with legs

9

cm and

12

cm gives hypotenuse

AB = 15

cm. Applying the Pythagorean theorem to right triangle

BPC

with legs

16

cm and

12

cm gives hypotenuse

BC = 20

cm. Since a kite has two pairs of equal adjacent sides (

AB = AD = 15

cm and

BC = CD = 20

cm), the total perimeter is

15 + 15 + 20 + 20 = 70

cm.

Step-by-Step Solution

1

Identify right triangles formed by the perpendicular diagonals

Four right triangles are formed:

\triangle APB

,

\triangle BPC

,

\triangle CPD

, and

\triangle DPA

.

Diagonals of a kite are perpendicular to each other.

2

Calculate upper side length

AB

AB = \sqrt{9^2 + 12^2} = \sqrt{225} = 15

cm

Apply the Pythagorean theorem

a^2 + b^2 = c^2

to right triangle

APB

.

3

Calculate lower side length

BC

BC = \sqrt{16^2 + 12^2} = \sqrt{400} = 20

cm

Apply the Pythagorean theorem

a^2 + b^2 = c^2

to right triangle

BPC

.

4

Compute the total perimeter

Perimeter

= 2(15) + 2(20) = 30 + 40 = 70

cm

A kite has two pairs of congruent adjacent sides (

AD = AB

and

CD = BC

).

Key Concept

Perpendicular diagonals and side length properties of a kite

Alternative Method

Instead of calculating all four sides individually, calculate one side from each distinct right triangle (

15

cm and

20

cm) and multiply their sum by

2

, using the property that a kite has two symmetric pairs of congruent adjacent sides:

2 \times (15 + 20) = 70

cm.

Estimated Time:1m 15s

Question 28Question →

In isosceles trapezoid $ABCD$ , side $AB$ is parallel to side $CD$ . If the measure of $\angle A$ is $(3x + 10)^\circ$ and the measure of $\angle C$ is $(5x - 30)^\circ$ , what is the measure of $\angle B$ ?

70^\circ

75^\circ

85^\circ

95^\circ

110^\circ

Show answer & explanation

Answer: $85^\circ$

Answer

The measure of

\angle B

is

85^\circ

.

The answer of

85^\circ

is correct because parallel sides

AB

and

CD

imply that consecutive interior angles

\angle A

and

\angle C

add up to

180^\circ

. Solving

(3x + 10) + (5x - 30) = 180

yields

x = 25

. Substituting

x = 25

into the expression for

\angle A

gives

3(25) + 10 = 85^\circ

. Because

ABCD

is an isosceles trapezoid, the base angles

\angle A

and

\angle B

adjacent to base

AB

are congruent, so

\angle B = 85^\circ

.

Step-by-Step Solution

1

Identify the relationship between

\angle A

and

\angle C

Since

AB \parallel CD

, angles

\angle A

and

\angle C

are consecutive interior angles along transversal

AC

(or leg

AD

), which means they are supplementary:

\angle A + \angle C = 180^\circ

.

Parallel lines cut by a transversal form supplementary consecutive interior angles.

2

Set up and solve the algebraic equation for

x

(3x + 10) + (5x - 30) = 180 \implies 8x - 20 = 180 \implies 8x = 200 \implies x = 25

.

Combine like terms and solve for

x

.

3

Calculate the measure of

\angle A

\angle A = 3(25) + 10 = 75 + 10 = 85^\circ

.

Substitute

x = 25

back into the expression for

\angle A

.

4

Determine the measure of

\angle B

using isosceles trapezoid properties

\angle B = \angle A = 85^\circ

.

In an isosceles trapezoid with

AB \parallel CD

, base angles along the same parallel base are congruent.

Key Concept

Properties of Isosceles Trapezoids and Consecutive Interior Angles

Question 29Question →

Determine whether the following statement is true or false:

The quadrilateral formed by connecting the midpoints of the four consecutive sides of any rhombus is always a rectangle.

True

False

Show answer & explanation

Answer: True

Answer

The statement is true because the diagonals of a rhombus are perpendicular, which forces the adjacent sides of the midpoint quadrilateral (which are parallel to the diagonals) to meet at right angles.

The statement is true because the diagonals of any rhombus are perpendicular. By the Midsegment Theorem, the sides of the quadrilateral formed by connecting adjacent midpoints are parallel to these diagonals, ensuring all four interior angles are

90^\circ

, which satisfies the definition of a rectangle.

Step-by-Step Solution

1

Apply the Triangle Midsegment Theorem to rhombus

ABCD

with diagonals

AC

and

BD

.

Let

P, Q, R,

and

S

be the midpoints of sides

AB, BC, CD,

and

DA

respectively. Segment

PQ

is parallel to diagonal

AC

and has length

\frac{1}{2}AC

. Segment

QR

is parallel to diagonal

BD

and has length

\frac{1}{2}BD

.

A line segment connecting the midpoints of two sides of a triangle is parallel to the third side and half its length.

2

Use the geometric properties of a rhombus's diagonals.

In any rhombus, the diagonals

AC

and

BD

are perpendicular to each other (

AC \perp BD

).

Perpendicular diagonals are a key defining property of all rhombuses.

3

Determine the interior angle measures of quadrilateral

PQRS

.

Because

PQ \parallel AC

and

QR \parallel BD

, and

AC \perp BD

, the sides

PQ

and

QR

must be perpendicular (

PQ \perp QR

). Therefore, all four interior angles of quadrilateral

PQRS

are

90^\circ

.

If two lines are parallel to two mutually perpendicular lines, they are also mutually perpendicular.

4

Classify quadrilateral

PQRS

.

Quadrilateral

PQRS

has four right angles, so it is by definition a rectangle.

Any quadrilateral with four right angles is a rectangle.

Key Concept

Properties of Rhombus Diagonals and Midpoint Quadrilaterals

Estimated Time:1m 0s

Question 30Question →

Kite $WXYZ$ has perpendicular diagonals $WY$ and $XZ$ that intersect at point $M$ . Diagonal $WY$ bisects diagonal $XZ$ such that $XM = MZ = 8$ centimeters. If $WM = 6$ centimeters and $MY = 15$ centimeters, what is the perimeter of kite $WXYZ$ , in centimeters?

27

46

54

68

74

Show answer & explanation

Answer: 54

Answer

The perimeter of kite

WXYZ

is 54 centimeters.

The correct answer is 54. The perpendicular diagonals of a kite form four interior right triangles. Using the legs

WM = 6

cm and

XM = 8

cm, the upper side

WX

is

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

cm. Using the legs

MY = 15

cm and

XM = 8

cm, the lower side

YX

is

\sqrt{15^2 + 8^2} = 17

cm. Summing all four outer sides (

10 + 10 + 17 + 17

) gives a total perimeter of 54 cm.

Step-by-Step Solution

1

Identify key geometric properties of the kite's diagonals.

The diagonals

WY

and

XZ

are perpendicular at intersection point

M

, creating four right triangles inside the kite.

By definition, the diagonals of a kite are perpendicular to each other.

2

Calculate the lengths of the upper pair of equal sides (

WX

and

WZ

).

WX = \sqrt{WM^2 + XM^2} = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10\text{ cm}

. Since

WZ = WX

,

WZ = 10\text{ cm}

.

Apply the Pythagorean theorem to right triangle

WMX

with legs of length 6 cm and 8 cm.

3

Calculate the lengths of the lower pair of equal sides (

YX

and

YZ

).

YX = \sqrt{MY^2 + XM^2} = \sqrt{15^2 + 8^2} = \sqrt{225 + 64} = \sqrt{289} = 17\text{ cm}

. Since

YZ = YX

,

YZ = 17\text{ cm}

.

Apply the Pythagorean theorem to right triangle

YMX

with legs of length 15 cm and 8 cm.

4

Compute the total perimeter of kite

WXYZ

.

\text{Perimeter} = WX + WZ + YX + YZ = 10 + 10 + 17 + 17 = 54\text{ cm}

.

Sum the lengths of all four exterior sides.

Key Concept

Properties of Kite Diagonals and Pythagorean Theorem

Estimated Time:1m 15s

Question 31Question →

In parallelogram $ABCD$ , the diagonals $AC$ and $BD$ intersect at point $E$ . If $AE = 2x + 5$ , $EC = 5x - 7$ , $BE = 3y - 1$ , and $ED = y + 9$ , what is the length of diagonal $BD$ ?

Show answer & explanation

Answer: 28

Answer

The length of diagonal

BD

is 28.

Because the diagonals of a parallelogram bisect each other, the intersection point

E

divides diagonal

BD

into two equal segments (

BE = ED

). Equating the expressions gives

3y - 1 = y + 9

, which simplifies to

2y = 10

, so

y = 5

. Substituting

y = 5

into the expression for

BE

yields

BE = 14

. Since

BD

consists of

BE + ED

, the full length of diagonal

BD

is

14 + 14 = 28

.

Step-by-Step Solution

1

Apply the diagonal bisection property of parallelograms.

BE = ED

, so

3y - 1 = y + 9

.

The diagonals of any parallelogram bisect each other at their intersection point.

2

Solve the linear equation for

y

.

2y = 10 \implies y = 5

.

Subtract

y

from both sides and add 1 to both sides.

3

Calculate segment

BE

and total length

BD

.

BE = 3(5) - 1 = 14

, so

BD = 2 \times 14 = 28

.

Substitute

y = 5

into the segment length expression and double it for the full diagonal length.

Key Concept

Diagonals of a parallelogram bisect each other.

Question 32Question →

In parallelogram $PQRS$ , the measures of consecutive angles $\angle P$ and $\angle Q$ are $(4x + 15)^\circ$ and $(2x + 45)^\circ$ , respectively. What is the measure, in degrees, of $\angle R$ ?

20^\circ

75^\circ

85^\circ

95^\circ

105^\circ

Show answer & explanation

Answer: $95^\circ$

Answer

95^\circ

In any parallelogram, consecutive interior angles formed by parallel lines and a transversal are supplementary (sum to

180^\circ

). Adding the expressions for consecutive angles

\angle P

and

\angle Q

gives

(4x + 15) + (2x + 45) = 180^\circ

, which simplifies to

6x + 60 = 180^\circ

, giving

x = 20

. Substituting

x = 20

into the expression for

\angle P

yields

4(20) + 15 = 95^\circ

. Because opposite angles in a parallelogram are congruent,

\angle R

has the same measure as

\angle P

, which is

95^\circ

.

Step-by-Step Solution

1

Set up an equation using the consecutive angle property of parallelograms.

(4x + 15) + (2x + 45) = 180

Consecutive angles in any parallelogram are supplementary (their sum is

180^\circ

).

2

Solve the linear equation for

x

.

6x + 60 = 180 \implies 6x = 120 \implies x = 20

Combine like terms (

4x + 2x = 6x

and

15 + 45 = 60

) and isolate

x

.

3

Calculate the measure of

\angle P

.

m\angle P = 4(20) + 15 = 80 + 15 = 95^\circ$

Substitute

x = 20

into the expression given for

\angle P

.

4

Determine the measure of

\angle R

.

m\angle R = m\angle P = 95^\circ$

Opposite angles in a parallelogram are equal in measure.

Key Concept

Angle Relationships in Parallelograms

Estimated Time:1m 0s

Question 33Question →

Determine whether the following statement regarding quadrilateral properties is true or false: Any convex quadrilateral whose diagonals intersect at right angles must be a rhombus.

True

False

Show answer & explanation

Answer: False

Answer

The statement is false. Perpendicular diagonals alone are not sufficient to prove that a quadrilateral is a rhombus.

The statement is false because perpendicular diagonals are not a sufficient condition to classify a general quadrilateral as a rhombus. Other quadrilaterals, such as kites or general orthodiagonal quadrilaterals with unequal sides, also have diagonals that intersect at right angles.

Step-by-Step Solution

1

Recall the defining properties of a rhombus.

A rhombus is a parallelogram with four congruent sides. Its diagonals are perpendicular bisectors of each other.

To evaluate whether perpendicular diagonals alone guarantee a rhombus.

2

Analyze counterexamples of non-rhombus quadrilaterals with perpendicular diagonals.

A kite has diagonals that intersect at right angles, but only adjacent pairs of sides are congruent, not all four sides. Furthermore, a general quadrilateral can have perpendicular diagonals of arbitrary lengths that do not bisect each other, resulting in four sides of completely different lengths.

A single counterexample disproves a universal mathematical claim.

3

Determine the overall truth value of the statement.

Because perpendicular diagonals are a necessary property of rhombuses but not a sufficient condition for all quadrilaterals, the statement is false.

Concluding the analysis based on geometric counterexamples.

Key Concept

Necessary vs. Sufficient Conditions for Quadrilateral Classification

Question 34Question →

In rhombus $JKLM$ , diagonals $JL$ and $KM$ intersect at point $P$ . If $JP = 2x + 3$ , $PL = 4x - 5$ , and $m\angle JKL = 60^\circ$ , what is the length of side $JK$ ?

8

11

22

11\sqrt{3}

44

Show answer & explanation

Answer: $22$

Answer

The length of side

JK

is

22

.

Since the diagonals of a rhombus bisect each other, segment

JP

equals segment

PL

. Equating

2x + 3 = 4x - 5

yields

x = 4

, which gives

JP = 11

. The diagonals of a rhombus are perpendicular and bisect the vertex angles, creating right triangle

\triangle JPK

with

m\angle JPK = 90^\circ

and

m\angle JKP = 30^\circ

. In a

30^\circ

-

60^\circ

-

90^\circ

right triangle, the hypotenuse is twice the side opposite the

30^\circ

angle. Because

JP = 11

is opposite the

30^\circ

angle, side

JK = 2 \times 11 = 22

. Alternatively, since

\triangle JKL

is an isosceles triangle with a

60^\circ

vertex angle, it is equilateral, making

JK = JL = 22

.

Step-by-Step Solution

1

Use the diagonal bisecting property of a rhombus to set up an algebraic equation.

2x + 3 = 4x - 5 \implies 2x = 8 \implies x = 4

The diagonals of a rhombus bisect each other, so

JP = PL

.

2

Calculate the length of the half-diagonal segment

JP

.

JP = 2(4) + 3 = 11

Substitute

x = 4

into the expression for

JP

.

3

Determine the angle measures in right triangle

\triangle JPK

.

m\angle JPK = 90^\circ

and

m\angle JKP = \frac{1}{2}(60^\circ) = 30^\circ

The diagonals of a rhombus are perpendicular to each other and bisect the vertex angles.

4

Apply the

30^\circ

-

60^\circ

-

90^\circ

right triangle ratio or sine function to find hypotenuse

JK

.

\sin(30^\circ) = \frac{JP}{JK} \implies \frac{1}{2} = \frac{11}{JK} \implies JK = 22$

In right triangle

\triangle JPK

,

JP

is opposite the

30^\circ

angle, so the hypotenuse

JK

is twice the length of

JP

.

Key Concept

Properties of Rhombus Diagonals and Special Right Triangles

Estimated Time:1m 30s

Question 35Question →

Quadrilateral $ABCD$ has vertices $A(-2, 1)$ , $B(2, 4)$ , $C(5, 0)$ , and $D(1, -3)$ in the standard $(x,y)$ coordinate plane. What is the area, in square units, of quadrilateral $ABCD$ ?

12.5

15

20

25

50

Show answer & explanation

Answer: $25$

Answer

25

Using the distance formula

d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

, each side length of quadrilateral

ABCD

is calculated to be

5

units. Evaluating the slopes of adjacent sides shows that the slope of

AB

is

\frac{3}{4}

and the slope of

BC

is

-\frac{4}{3}

. Since their product is

-1

, adjacent sides are perpendicular. A quadrilateral with four equal sides and right angles is a square, and its area is

5^2 = 25

square units.

Step-by-Step Solution

1

Calculate the length of side

AB

using the distance formula.

AB = \sqrt{(2 - (-2))^2 + (4 - 1)^2} = \sqrt{4^2 + 3^2} = \sqrt{25} = 5

.

The distance formula finds the exact side length of the quadrilateral.

2

Calculate adjacent side length

BC

and check the slopes to determine the figure type.

BC = \sqrt{(5 - 2)^2 + (0 - 4)^2} = \sqrt{3^2 + (-4)^2} = 5

. The slope of

AB

is

\frac{3}{4}

and the slope of

BC

is

-\frac{4}{3}

.

Because all sides are equal to

5

and adjacent slopes are negative reciprocals, the figure is a square.

3

Calculate the area of the square.

\text{Area} = \text{side}^2 = 5^2 = 25

.

The area of a square with side length

s

is

s^2

.

Key Concept

Finding the area of a quadrilateral in the coordinate plane by verifying side lengths and right angles using distance and slope formulas.

Question 36Question →

In parallelogram $ABCD$ , diagonals $AC$ and $BD$ intersect at point $E$ . In the standard $(x,y)$ coordinate plane, vertex $A$ is located at $(-3, 2)$ and point $E$ is located at $(2, 1)$ . What are the coordinates of vertex $C$ ?

(-0.5, 1.5)

(3, -5)

(5, -1)

(7, 0)

(7, 4)

Show answer & explanation

Answer: $(7, 0)$

Answer

The coordinates of vertex

C

are

(7, 0)

.

A fundamental property of any parallelogram is that its diagonals bisect each other. Therefore, the intersection point

E(2, 1)

must be the midpoint of diagonal

AC

. Using the midpoint formula,

\frac{-3 + x_C}{2} = 2

gives

x_C = 7

, and

\frac{2 + y_C}{2} = 1

gives

y_C = 0

. Thus, the coordinates of vertex

C

are

(7, 0)

.

Step-by-Step Solution

1

Identify the key geometric property of parallelograms.

The diagonals of a parallelogram bisect each other, which means point

E

is the midpoint of diagonal

AC

.

By definition of diagonal bisection in any parallelogram, the intersection point of the diagonals is the midpoint of both diagonal segments.

2

Set up the midpoint formula for segment

AC

with midpoint

E(2, 1)

.

(\frac{x_A + x_C}{2}, \frac{y_A + y_C}{2}) = (2, 1)

, where

x_A = -3

and

y_A = 2

.

The midpoint coordinates are the averages of the endpoint coordinates.

3

Solve for the

x

-coordinate of vertex

C

.

\frac{-3 + x_C}{2} = 2 \implies -3 + x_C = 4 \implies x_C = 7

.

Multiply by 2 and add 3 to isolate

x_C

.

4

Solve for the

y

-coordinate of vertex

C

.

\frac{2 + y_C}{2} = 1 \implies 2 + y_C = 2 \implies y_C = 0

.

Multiply by 2 and subtract 2 to isolate

y_C

.

Key Concept

Diagonals of a Parallelogram Bisect Each Other

Alternative Method

Use vector translations: The vector from

A(-3, 2)

to

E(2, 1)

is

\langle 2 - (-3), 1 - 2 \rangle = \langle 5, -1 \rangle

. Since

E

is the midpoint of

AC

, the vector from

E

to

C

is identical. Adding

\langle 5, -1 \rangle

to

E(2, 1)

gives

C(2 + 5, 1 - 1) = (7, 0)

.

Estimated Time:1m 15s

Question 37Question →

In any trapezoid, the length of the midsegment connecting the midpoints of the non-parallel legs is equal to half the difference of the lengths of the two parallel bases.

True

False

Show answer & explanation

Answer: False

Answer

False. The length of the midsegment of a trapezoid is equal to half the sum of the lengths of the parallel bases, not half the difference.

The statement is false because the Trapezoid Midsegment Theorem dictates that the length of the midsegment connecting the midpoints of the non-parallel legs is equal to half the sum (the average) of the lengths of the parallel bases,

\frac{b_1 + b_2}{2}

. Half the difference of the base lengths gives the length of the segment connecting the midpoints of the diagonals.

Step-by-Step Solution

1

Recall the definition of a trapezoid midsegment.

The midsegment of a trapezoid is the line segment joining the midpoints of its non-parallel legs.

Identifying the geometric figure and segment being described is necessary to apply the correct theorem.

2

Apply the Trapezoid Midsegment Theorem.

The length

m

of the midsegment for a trapezoid with parallel bases

b_1

and

b_2

is given by

m = \frac{b_1 + b_2}{2}

.

The theorem states that the midsegment length is the average (half the sum) of the two base lengths.

3

Compare the theorem formula to the given statement.

The statement claims the length is

\frac{|b_1 - b_2|}{2}

(half the difference), which contradicts the true formula

\frac{b_1 + b_2}{2}

. Therefore, the statement is false.

Evaluating the mathematical validity of the statement determines the correct answer.

Key Concept

Trapezoid Midsegment Theorem

Question 38Question →

In rectangle $ABCD$ , diagonals $AC$ and $BD$ intersect at point $E$ . If the measure of $\angle AEB$ is $120^\circ$ and $AC = 16$ , what is the length of side $BC$ ?

Show answer & explanation

Answer: 8

Answer

The length of side

BC

is

8

.

In any rectangle, the diagonals are congruent and bisect each other. Given

AC = 16

, the distance from the intersection point

E

to any vertex is

8

, so

BE = EC = 8

. Because

\angle AEB

and

\angle BEC

lie along the straight diagonal line

AC

, they are supplementary, giving

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

. Triangle

BEC

is an isosceles triangle with

BE = EC = 8

and a vertex angle of

60^\circ

, which forces it to be equilateral. Consequently, all sides of

\triangle BEC

are equal, so

BC = 8

.

Step-by-Step Solution

1

Calculate the lengths of the diagonal segments from the center point

E

.

BE = EC = 8

The diagonals of a rectangle are congruent and bisect each other, so each half-diagonal equals half of

AC

.

2

Determine the measure of adjacent angle

\angle BEC

.

\angle BEC = 60^\circ

Angles

\angle AEB

and

\angle BEC

form a straight line (linear pair), so their sum is

180^\circ

.

3

Analyze

\triangle BEC

to find the length of side

BC

.

BC = 8

An isosceles triangle with a

60^\circ

vertex angle has base angles of

60^\circ

as well, making it an equilateral triangle where all sides are equal to

8

.

Key Concept

Diagonals of a rectangle are equal in length and bisect each other, dividing the rectangle into two pairs of congruent isosceles triangles.

Estimated Time:1m 15s

Question 39Question →

A rhombus has a perimeter of 40 centimeters and one diagonal of length 12 centimeters. What is the area, in square centimeters, of the rhombus?

48

60

120

192

96

Show answer & explanation

Answer: 96

Answer

96 square centimeters

Because all four sides of a rhombus are congruent, a perimeter of 40 centimeters gives a side length of 10 centimeters. The diagonals of a rhombus intersect at right angles and bisect each other. Half of the given diagonal is 6 centimeters. By applying the Pythagorean theorem (

6^2 + b^2 = 10^2

), the second half-diagonal is found to be 8 centimeters, which means the full length of the second diagonal is 16 centimeters. Using the rhombus area formula

\text{Area} = \frac{1}{2} \cdot d_1 \cdot d_2

, the area is

\frac{1}{2} \cdot 12 \cdot 16 = 96

square centimeters.

Step-by-Step Solution

1

Calculate the side length of the rhombus from its perimeter.

Side length

s = \frac{40}{4} = 10\text{ cm}

.

All four sides of a rhombus are equal in length.

2

Use the properties of rhombus diagonals to find the length of the second diagonal.

Half of the given diagonal is

\frac{12}{2} = 6\text{ cm}

. In the right triangle formed by the half-diagonals and a side:

6^2 + b^2 = 10^2 \implies 36 + b^2 = 100 \implies b = 8\text{ cm}

. Thus, the second diagonal

d_2 = 2 \times 8 = 16\text{ cm}

.

The diagonals of a rhombus are perpendicular bisectors of each other.

3

Calculate the area of the rhombus using the diagonal formula.

\text{Area} = \frac{1}{2} \times d_1 \times d_2 = \frac{1}{2} \times 12 \times 16 = 96\text{ sq cm}.

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

Key Concept

Properties of Rhombus Diagonals and Area Calculation

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