Question 280·Medium·Math

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Triangle $ABC$ is similar to triangle $DEF$ , with $A$ , $B$ , and $C$ corresponding to $D$ , $E$ , and $F$ , respectively. In triangle $ABC$ , $AB = 6$ and $BC = 9$ . In triangle $DEF$ , $DE = 10$ .

What is the length of $\overline{EF}$ ?

If triangles are similar, their matching sides share one scale factor. The stated vertex order tells you which sides match, so pair them by their endpoints. Write second-triangle length over first-triangle length for both pairs, then use a Desmos regression to find the missing side. Two sides of one triangle don't give the scale between triangles.

Hints

Hint 1
Corresponding vertices are the ones paired in order: $A$ with $D$ , $B$ with $E$ , and $C$ with $F$ . A side matches the side joining its paired endpoints. Which side matches $BC$ ?
Hint 2
A scale factor multiplies every side of one triangle to get its matching side in the other. Compare $DE$ with $AB$ , then keep that same direction when you compare $EF$ with $BC$ .

Step-by-step

Match sides and solve a proportion
Step 1
Pair the corresponding sides
Corresponding means matching. The stated order pairs $A$ with $D$ , $B$ with $E$ , and $C$ with $F$ . A side joins two vertices, so match its endpoints: $AB$ matches $DE$ , and $BC$ matches $EF$ .
Step 2
Write the matching side ratios
Let $x=EF$ . Similar triangles use the same scale factor for each pair of matching sides. Keep second-triangle length over first-triangle length in both ratios:
$\frac{DE}{AB}=\frac{EF}{BC}$
Substitute the given lengths:
$\frac{10}{6}=\frac{x}{9}$
Comparing $9$ with $6$ would use two sides of $ABC$ , not a scale between triangles.
Step 3
Find the missing side in Desmos
Type $\frac{10}{6}\sim\frac{x_1}{9}$ . Using $x_1$ instead of $x$ and $\sim$ instead of $=$ makes this a regression: Desmos finds the number that makes the sides equal. Under PARAMETERS, it shows $x_1=15$ . Since $x_1$ represents $EF$ , the requested length is $15$ . Choice C.

Lessons that teach this

SAT Geometry and TrigonometryIntermediateCoreSolve similar-triangle proportions

Question 280·Medium·Math

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