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

Q: Why do I need to use the Pythagorean theorem here?

The height and base difference form right triangles! When you drop perpendiculars from the top vertices, you create right triangles with height = 10 cm and base = 2 cm (half the difference between bases).

Q: How do I find the base of the right triangle?

Take the difference between the parallel sides: $ 19 - 15 = 4 $ cm. Since it's an isosceles trapezoid, this difference is split equally on both sides, so each base = 2 cm.

Q: Can I simplify √104 further?

Yes! $ \sqrt{104} = \sqrt{4 \times 26} = 2\sqrt{26} $. So the perimeter can also be written as 34 + 4√26 cm.

Q: What if the trapezoid wasn't isosceles?

Then the two non-parallel sides would have different lengths, and you'd need additional information to solve the problem. The isosceles property is crucial here!

Q: Why can't I just use the formula for trapezoid area?

Area formulas won't help find perimeter! Perimeter is the distance around the outside, so you need the length of each side. Area and perimeter are completely different measurements.

Isosceles Trapezoid Perimeter with Pythagorean Applications

The height of the isosceles trapezoid ABCD is 10 cm.

AB = 15 cm

CD = 19 cm

Calculate the perimeter of the trapezoid.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

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 written solution

Follow each step carefully to understand the complete solution

Understand the problem

The height of the isosceles trapezoid ABCD is 10 cm.

AB = 15 cm

CD = 19 cm

Calculate the perimeter of the trapezoid.

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 $34 + 2\sqrt{104} \, \text{cm}$ .

Final Answer

$34+2\sqrt{104}$ cm

Key Points to Remember

Essential concepts to master this topic

Rule: In isosceles trapezoids, both non-parallel sides are equal
Technique: Use Pythagorean theorem: $AD^2 = 10^2 + 2^2 = 104$
Check: Verify legs equal: $AD = BC = \sqrt{104}$ cm ✓

Common Mistakes

Avoid these frequent errors

Adding base lengths directly without finding leg lengths
Don't just add AB + CD = 15 + 19 = 34 cm as the perimeter! This ignores the slanted sides completely and gives an incomplete answer. Always use the Pythagorean theorem to find the leg lengths first, then add all four sides.

Practice Quiz

Test your knowledge with interactive questions

Given the trapezoid:

What is the area?

FAQ

Everything you need to know about this question

Why do I need to use the Pythagorean theorem here?

The height and base difference form right triangles! When you drop perpendiculars from the top vertices, you create right triangles with height = 10 cm and base = 2 cm (half the difference between bases).

How do I find the base of the right triangle?

Take the difference between the parallel sides: $19 - 15 = 4$ cm. Since it's an isosceles trapezoid, this difference is split equally on both sides, so each base = 2 cm.

Can I simplify √104 further?

Yes! $\sqrt{104} = \sqrt{4 \times 26} = 2\sqrt{26}$ . So the perimeter can also be written as 34 + 4√26 cm.

What if the trapezoid wasn't isosceles?

Then the two non-parallel sides would have different lengths, and you'd need additional information to solve the problem. The isosceles property is crucial here!

Why can't I just use the formula for trapezoid area?

Area formulas won't help find perimeter! Perimeter is the distance around the outside, so you need the length of each side. Area and perimeter are completely different measurements.

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