Perimeter of an Isosceles Trapezoid
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In the figure shown, $\overline{BC}$ is parallel to $\overline{AD}$ and $AB = CD$. What is the perimeter of quadrilateral $ABCD$?
This question includes visual content: A diagram shows an isosceles trapezoid ABCD. The top base BC has a length of 10. The bottom base AD has a length of 26. A dashed vertical line representing height is drawn from the top base to the bottom base, forming a right angle at the bottom base; this height is labeled 6. The non-parallel sides AB and CD are shown as congruent through symmetry and problem text.
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In this problem, we need to calculate the perimeter of an isosceles trapezoid using the given side lengths and height.
Perimeter of Quadrilateral ABCD
Let's list what we know. The top base B C is ten, the bottom base A D is twenty-six, the height is six, and side AB equals side CD, which makes it an isosceles trapezoid.
| Given Information | Value |
|---|---|
| Top Base ($b_1$) | 10 |
| Bottom Base ($b_2$) | 26 |
| Height ($h$) | 6 |
To find the perimeter, we need the lengths of the non-parallel sides. Let's redraw the figure and drop two altitudes from vertices B and C to the base A D.
The central part of the base is equal to the top base, which is ten.
Because the trapezoid is isosceles, the remaining length of the bottom base, twenty-six minus ten, is split equally between the two outer segments.
Dividing sixteen by two gives us eight for each of the small segments on the left and right.
Now we have a right-angled triangle with a base of eight and a height of six. We can use the Pythagorean theorem to find the length of the slant side s.
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