Isosceles Trapezoid: Finding the Height from Its Sides

Grades 8-9

Question: Draw a trapezoid ABCD, with AB and CD as the bases. Make AD = DC = CB = 5 and AB = 10.

To construct a trapezoid with the given dimensions (

AD = DC = CB = 5

and

AB = 10

), we first need to identify its geometric properties. Since the non-parallel sides

AD

and

BC

are equal, this is an isosceles trapezoid.

To determine the exact position of the top vertices (

C

and

D

), we can follow this reasoning:

Computing the base and height: If we drop perpendiculars from $D$ and $C$ to the longer base $AB$ , call the intersection points $D'$ and $C'$ . Since the trapezoid is symmetric, segment $AD'$ equals segment $C'B$ . Since the top base $CD$ measures $5$ and the bottom base $AB$ measures $10$ , the leftover space along the longer base is $10 - 5 = 5$ . This space splits evenly between the two sides: $\begin{align*} AD' = C'B = \frac{10 - 5}{2} = 2.5 \end{align*}$
Using the Pythagorean Theorem: In right triangle $\triangle AD'D$ , we know the hypotenuse ( $AD = 5$ ) and one leg ( $AD' = 2.5$ ). The trapezoid's height ( $h$ ) is: $\begin{align*} h^2 + (2.5)^2 &= 5^2 \\ h^2 + 6.25 &= 25 \\ h^2 &= 18.75 \\ h &= \sqrt{18.75} \approx 4.33 \end{align*}$

In the diagram, the trapezoid is highlighted in blue. The labels show that the bottom base is twice the length of the top base, and the three remaining sides are identical, forming a perfectly symmetric figure.