Look at the rectangle ABC is below. AB = 4 AD = 3 Determine the length of the diagonal AC? 4 4 4 3 3 3 A A A B B B C C C D D D M M M

It is not possible to calculate

Rectangle Diagonals Practice Problems - Geometry Worksheets

Master rectangle diagonal properties with step-by-step practice problems. Learn how diagonals bisect each other, prove triangle congruence, and solve geometry exercises.

📚Practice Rectangle Diagonal Properties and Proofs

Prove that rectangle diagonals are equal and bisect each other
Calculate areas of triangles formed by intersecting diagonals
Use SAS postulate to prove triangle congruence in rectangles
Determine angle relationships when diagonals intersect
Apply diagonal properties to solve real-world geometry problems
Master vertical angles and alternate angles in rectangle proofs

Understanding Diagonals

Complete explanation with examples

Diagonals in a Rectangle

Properties of rectangle diagonals:

• The diagonals of a rectangle are equal to each other.
• The diagonals of a rectangle bisect each other.
• The diagonals of a rectangle are not perpendicular to each other.
• The diagonals of a rectangle are not angle bisectors.

Detailed explanation

Practice Diagonals

Test your knowledge with 6 quizzes

Given the rectangle ABCD

It is known that:

AB=4

AD=3

What is the length of the diagonal BD?

Examples with solutions for Diagonals

Step-by-step solutions included

Exercise #1

The rectangle ABCD is shown below.

DC = 24

BM = 12.5

Which segments are equal to BM?

Step-by-Step Solution

It is important to remember that the diagonals in a rectangle intersect and are equal to each other,

therefore:

$BM=MD=AM=MC=12.5$

Answer:

BM=MD=AM=MC=12.5

Video Solution

Exercise #2

The rectangle ABCD is shown below.

Angle CAD is equal to 45 degrees.

Calculate the remaining angles in the rectangle.

Step-by-Step Solution

Let's observe triangle CAD, the sum of angles in a triangle is 180 degrees, hence we can determine angle DAC:

$CAD+90+30=180$

$CAD+120=180$

$CAD=180-120$

$CAD=60$

Given that ABCD is a rectangle, all angles are equal to 90 degrees.

Therefore angle CAB equals:

$90-CAD=90-60=30$

Furthermore we can deduce that CAD equals 30 degrees, since ABCD is a rectangle all angles are equal to 90 degrees.

CAB equals 60 degrees.

Therefore:

$CAD=BCA=30,ACD=CAB=60$

Answer:

CAD = BCA = 30
ACD = CAB = 60

Video Solution

Exercise #3

Calculate the perimeter of the rectangle ABCD.

Step-by-Step Solution

Let's focus on triangle BCD in order to find side BC.

We'll use the Pythagorean theorem using our values:

$BC^2+DC^2=BD^2$

$BC^2+24^2=25^2$

$BC^2=625-576=49$

Let's now remove the square root:

$BC=7$

Since each pair of opposite sides are equal to each other in a rectangle, we can state that:

$DC=AB=24$

$BC=AD=7$

Now we can calculate the perimeter of the rectangle by adding all sides together:

$24+7+24+7=14+48=62$

Answer:

Video Solution

Exercise #4

Look at the following rectangle:

Calculate the perimeter of the rectangle ABCD.

Step-by-Step Solution

Let's focus on triangle BCD in order to find side DC.

We'll use the Pythagorean theorem and input the known data:

$BC^2+DC^2=BD^2$

$6^2+DC^2=10^2$

$DC^2=100-36=64$

Let's now remove the square root:

$DC=8$

Since in a rectangle each pair of opposite sides are equal to each other, we know that:

$DC=AB=8$

$BC=AD=6$

Now we can calculate the perimeter of the rectangle by adding all sides together:

$8+6+8+6=16+12=28$

Answer:

Video Solution

Exercise #5

Look at the following rectangle:

AB = 12

AC = 13

Calculate the area of the triangle BCD.

Step-by-Step Solution

To solve this problem, we'll follow these steps:

Step 1: Use the Pythagorean Theorem to calculate the length of $AD$ .
Step 2: Calculate the area of triangle $\triangle BCD$ .

Step 1: Given $AB = 12$ and $AC = 13$ , we use the Pythagorean Theorem to find $AD$ .

AC^2 = AB^2 + AD^2 \implies 13^2 = 12^2 + AD^2

169 = 144 + AD^2 \implies AD^2 = 25 \implies AD = 5

Step 2: Knowing the sides $AD = 5$ (height of the rectangle) and $AB = 12$ (base of the rectangle), triangle $\triangle BCD$ will have the base $BC = 12$ and the height $BD = 5$ .

The area of triangle $\triangle BCD$ is:

\text{Area}_{\triangle BCD} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 12 \times 5 = 30

Therefore, the area of triangle $\triangle BCD$ is 30.

Answer:

Video Solution

Frequently Asked Questions

Everything you need to know about Diagonals

What are the main properties of diagonals in a rectangle?

Rectangle diagonals have four key properties: they are equal in length (AC = BD), they bisect each other at the intersection point, they are not perpendicular to each other, and they do not bisect the rectangle's angles. These properties are fundamental for solving rectangle geometry problems.

How do you prove triangles are congruent when rectangle diagonals intersect?

Use the Side-Angle-Side (SAS) postulate. Since diagonals bisect each other, corresponding segments are equal. Vertical angles at the intersection are equal. With two equal sides and the included angle, triangles like AEB and DEC are congruent.

Why are all four triangles formed by rectangle diagonals equal in area?

The four triangles have equal areas because: 1) Diagonals bisect each other, creating equal bases, 2) They share the same height from the intersection point, 3) Congruent triangles have identical areas. This follows from the area formula: (base × height) ÷ 2.

Are rectangle diagonals perpendicular to each other?

No, rectangle diagonals are not perpendicular unless the rectangle is a square. In a regular rectangle, the diagonals intersect at an angle that is not 90 degrees. Only in squares do the diagonals meet at right angles.

How do you calculate the length of a rectangle diagonal?

Use the Pythagorean theorem: diagonal = √(length² + width²). For example, if a rectangle has length 6 and width 8, the diagonal length is √(6² + 8²) = √(36 + 64) = √100 = 10 units.

What is the difference between rectangle and parallelogram diagonals?

Rectangle diagonals are always equal in length and bisect each other, while general parallelogram diagonals only bisect each other but are not necessarily equal. Rectangles are special parallelograms with additional properties due to their right angles.

How do rectangle diagonal properties help solve geometry proofs?

Diagonal properties provide multiple proof strategies: equal diagonal lengths for establishing congruence, bisection property for finding equal segments, and the four equal triangles for area calculations. These properties are essential tools in geometric reasoning and problem-solving.

Can rectangle diagonals be angle bisectors?

No, rectangle diagonals do not bisect the rectangle's angles unless the rectangle is a square. In a regular rectangle, the diagonals create different angles on each side, so angles like ABE and CBE are not equal.

Continue Your Math Journey

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Rectangle Diagonals Practice Problems - Geometry Worksheets

Understanding Diagonals

Diagonals in a Rectangle

Properties of rectangle diagonals:

Practice Diagonals

Examples with solutions for Diagonals

Frequently Asked Questions

What are the main properties of diagonals in a rectangle?

How do you prove triangles are congruent when rectangle diagonals intersect?

Why are all four triangles formed by rectangle diagonals equal in area?

Are rectangle diagonals perpendicular to each other?

How do you calculate the length of a rectangle diagonal?

What is the difference between rectangle and parallelogram diagonals?

How do rectangle diagonal properties help solve geometry proofs?

Can rectangle diagonals be angle bisectors?

Continue Your Math Journey

Suggested Topics to Practice in Advance

Topics Learned in Later Sections

Practice by Question Type