Calculate Rectangle Perimeter with Isosceles Triangle: 4 and 8 Unit Problem

Look at the following rectangle:

ΔAEB is isosceles (AE=EB).

Calculate the perimeter of the rectangle ABCD.

Video Solution

Solution Steps

00:00 Determine the perimeter of the rectangle

00:06 In a rectangle all angles are right angles

00:09 Apply the Pythagorean theorem in triangle ADE

00:12 Insert the appropriate values into the expression and solve to find DE

00:19 Isolate DE

00:30 This is the length of DE

00:42 According to the given data this is an Isosceles triangle

00:48 Opposite sides are equal in a rectangle

00:52 Right angles in a rectangle

00:59 Therefore the triangles are congruent by AAS

01:05 The sides are equal because the triangles are congruent

01:16 The side equals the sum of its parts

01:27 The perimeter of the rectangle equals the sum of its sides

01:37 Let's group the terms together

01:47 Break down 48 into factors of 16 and 3

01:52 Solve the square root of 16

01:56 This is the solution

Step-by-Step Solution

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

Now, let's work through each step:
Step 1: For triangle $ACD$ , where $AC = 8$ and $AD = 4$ , apply the Pythagorean theorem:

AC^2 = AB^2 + AD^2

8^2 = AB^2 + 4^2

64 = AB^2 + 16

AB^2 = 64 - 16 = 48

AB = \sqrt{48} = 4\sqrt{3}

Step 2: The perimeter of rectangle $ABCD$ is given by:

Perimeter = 2 \times (AB + AD) = 2 \times (4\sqrt{3} + 4)

Perimeter = 2 \times 4(1 + \sqrt{3}) = 8 + 8\sqrt{3}

Therefore, the solution to the problem is $8 + 16\sqrt{3}$ .