Question 292·Medium·Math

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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 $\overline{EC}$ ?

In a trapezoid with parallel bases and crossing diagonals, similar triangles form above and below the crossing. Parallel lines give one pair of equal angles, and the intersection gives another. Match the bases and diagonal pieces, then solve their proportion with a Desmos regression. Reversing only one ratio makes the requested piece too short.

Hints

Hint 1
Look at the triangle above $E$ and the one below it. Their angles at $E$ are vertical angles, which means they sit opposite each other where two lines cross. What other equal angles come from the parallel bases?
Hint 2
For similar triangles, matching sides grow by the same factor. The short base matches the long base; the piece of diagonal from $A$ to $E$ matches the piece from $E$ to $C$ . Keep both ratios in the same direction.

Step-by-step

Match the triangles and solve a proportion
Step 1
Show why the triangles match
At $E$ , $\angle AEB$ and $\angle CED$ are vertical angles, so they're equal. Diagonal $AC$ crosses the parallel bases, so $\angle BAE$ and $\angle DCE$ are equal too. Two equal angle pairs give AA similarity: $\triangle ABE\sim\triangle CDE$ . The order shows that $A$ matches $C$ , $B$ matches $D$ , and $E$ matches $E$ .
Step 2
Write the matching-side proportion
Match sides by their endpoints before you write the ratio. $AB$ matches $CD$ , and $AE$ matches $CE$ . Let $x_1$ be $EC$ . The lower triangle's side goes on top in both ratios, so $\frac{x_1}{7.2}=\frac{18}{8}$ . Don't flip only the diagonal ratio: the longer base belongs to the triangle with $EC$ .
Step 3
Find the diagonal piece
Type $\frac{x_1}{7.2}\sim\frac{18}{8}$ in Desmos. Use $\sim$ to tell Desmos to find the value of $x_1$ that makes the sides equal. Under PARAMETERS, it shows $x_1=16.2$ . So the length of $EC$ is $16.2$ . Choice C.

Question 292·Medium·Math

Strategy

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Step-by-step

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Strategy

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