Question 90·Medium·Geometry and Trigonometry

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In trapezoid $ABCD$ , $\overline{AB}$ is parallel to $\overline{CD}$ . Diagonals $\overline{AC}$ and $\overline{BD}$ intersect at point $E$ . If $AB=8$ , $CD=18$ , and $AE=7.2$ , what is the length of $\overline{EC}$ ?

When a trapezoid has parallel bases and intersecting diagonals, look for pairs of triangles that share the intersection point and have angles formed by a transversal across parallel lines. Prove the triangles are similar using a vertical-angle pair and an angle pair created by the parallel lines, then write a proportion matching corresponding sides (base to base and diagonal segment to diagonal segment) and solve.

Hints

Click me!

Look for triangles created by the diagonals

The diagonals intersect at $E$ , forming several triangles. Focus on one triangle that uses $\overline{AB}$ and one that uses $\overline{CD}$ .

Use the parallel lines to match angles

Because $\overline{AB} \parallel \overline{CD}$ , angles formed with the diagonal $\overline{BD}$ can be matched as equal angles.

Set up a ratio on diagonal $AC$

Once you identify similar triangles, relate the segments on diagonal $AC$ (the pieces $AE$ and $EC$ ) to the parallel bases $AB$ and $CD$ .

Desmos Guide

Enter two expressions that should be equal

Use the proportion $\frac{7.2}{x}=\frac{8}{18}$ . In Desmos, graph:

$y=7.2/x$
$y=8/18$

Find the intersection

Click the intersection point of the two graphs. The $x$ -coordinate is the value that makes the proportion true.

Interpret the result

The $x$ -coordinate from the intersection represents the length of $\overline{EC}$ .

Step-by-step Explanation

Show two triangles are similar

Consider $\triangle ABE$ and $\triangle CDE$ .

$\angle AEB$ and $\angle CED$ are vertical angles, so they are equal.
$\angle ABE$ equals $\angle CDE$ because $\overline{AB} \parallel \overline{CD}$ and $\overline{BD}$ is a transversal.

Therefore, $\triangle ABE \sim \triangle CDE$ .

Write a proportion with corresponding sides

From the similarity, corresponding sides are proportional:

\frac{AE}{EC}=\frac{AB}{CD}

Substitute the given values:

\frac{7.2}{EC}=\frac{8}{18}

Solve for $EC$

Cross-multiply:

7.2\cdot 18 = 8\cdot EC

EC=\frac{7.2\cdot 18}{8}=16.2

Therefore, the length of $\overline{EC}$ is 16.2.

Question 90·Medium·Geometry and Trigonometry

Strategy

Hints

Desmos Guide

Step-by-step Explanation

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Strategy

Hints

Desmos Guide

Step-by-step Explanation