Calculate Isosceles Trapezoid Perimeter: 15cm Top, 19cm Bottom, 10cm Height

The height of the isosceles trapezoid ABCD is 10 cm.

AB = 15 cm

CD = 19 cm

Calculate the perimeter of the trapezoid.

Video Solution

Solution Steps

00:00 Find the perimeter of the trapezoid

00:03 Draw the second height to create a rectangle

00:10 Opposite sides are equal in the rectangle

00:14 Overlapping sides in overlapping triangles

00:17 Calculate the remainder of CD minus FE

00:26 The length of each segment equals half of what remains from side DC

00:34 Now use the Pythagorean theorem in triangle AFC

00:39 Substitute appropriate values and solve to find leg AC

00:50 This is the length of leg AC, and also BD since the trapezoid has equal legs

01:10 The perimeter of the trapezoid equals the sum of its sides

01:18 Substitute appropriate values and solve to find the perimeter

01:31 And this is the solution to the problem

Step-by-Step Solution

To find the perimeter of the isosceles trapezoid ABCD, we'll follow these steps:

Step 1: Identify the parameters: - Top base $AB = 15$ cm. - Bottom base $CD = 19$ cm. - Height $h = 10$ cm.
Step 2: Calculate the length of the non-parallel sides (legs). - The difference between the bases is $CD - AB = 19 - 15 = 4$ cm. - Divide this difference symmetrically: each segment between the parallel bases (on the x-axis) is $2$ cm long. - Right triangle segment formed with height and half-difference: height = 10 cm, base = 2 cm.
Step 3: Use the Pythagorean Theorem to calculate the leg length $AD$ (same as $BC$ ): $AD^2 = 10^2 + 2^2$ $AD^2 = 100 + 4 = 104$ $AD = \sqrt{104} \, \text{cm}$ Each non-parallel side $AD$ $= BC = \sqrt{104}$ cm.
Step 4: Calculate the perimeter: - Perimeter $= AB + CD + DA + BC$ $= 15 + 19 + \sqrt{104} + \sqrt{104}$ $= 34 + 2\sqrt{104} \, \text{cm}$

Therefore, the perimeter of the trapezoid is $34 + 2\sqrt{104} \, \text{cm}$ .