Question type

Similar triangles on the SAT

Similar triangles have equal corresponding angles and proportional corresponding sides. Match vertices in order, then set up one ratio: if $△ A B C \sim △ D E F$ , then $\frac{A B}{D E} = \frac{B C}{E F}$ . A segment parallel to one side of a triangle cuts off a smaller similar triangle, so $\frac{A D}{A B} = \frac{A E}{A C}$ .

These questions give two similar triangles, a segment drawn parallel to one side, two lines crossing between parallel segments, or a dilation, and ask for a missing side, a ratio or a perimeter.

It appeared on four of the five official practice tests we analysed, five times in all.

On 4 of 5 official practice tests
38 practice questions of this type
Desmos partly helps

Why students miss it

Matching the wrong sides. Correspondence follows the order of the letters, not the position on the page.

Using a piece instead of the whole side. With $D E ∥ B C$ , the ratio is $\frac{A D}{A B}$ , where $A B = A D + D B$ , not $\frac{A D}{D B}$ against $\frac{A E}{A C}$ .

Scaling areas by the length ratio. Perimeters scale by $k$ , areas by $k^{2}$ .

How to spot it

"Similar", the symbol $\sim$ , or "dilated by a scale factor".
A segment drawn parallel to one side of a triangle.
Two segments crossing at a point, with their ends joined by parallel segments.

The method

Set up the proportion

Match corresponding vertices, then write one ratio of corresponding sides.

Named similarity, $△ A B C \sim △ D E F$: $\frac{A B}{D E} = \frac{B C}{E F} = \frac{A C}{D F}$ .
Parallel segment inside a triangle: small side over whole side, on both sides.
Crossing segments between parallels: $\frac{A E}{E B} = \frac{C E}{E D}$ .
Dilation by $k$: every length, and the perimeter, times $k$ .

With Desmos

Desmos solves the proportion; it cannot tell you which sides correspond.

Type the proportion with x for the unknown, for example 4/10=6/x, and read x = 15.

Two real questions

From the 1500prep bank, the way the app shows them. Answer first, then open the explanation.

1Question 1 of 2#877Geo

In triangle ABC, point D lies on side AB and point E lies on side AC such that DE is parallel to BC. If AD = 4, DB = 6, and AE = 6, what is the length of EC?

Pick an answer, then check it.

Show the answer and explanation

D E ∥ B C

makes the small triangle similar to the big one, so the sides are cut proportionally:

\frac{A D}{D B} = \frac{A E}{E C}

. Then

\frac{4}{6} = \frac{6}{E C}

, so

4 \cdot E C = 36

and

E C = 9

2Question 2 of 2#2281Geo

Segments $A B$ and $C D$ intersect at point $E$ , and $A C$ is parallel to $B D$ . If $A E = 6$ , $E B = 9$ , and $C E = 8$ , what is the length of $E D$ ?

Pick an answer, then check it.

Show the answer and explanation

Vertical angles at

E

plus

A C ∥ B D

(alternate interior angles) make triangle

A E C \sim

triangle

B E D

, with

A

matching

B

and

C

matching

D

. So

\frac{A E}{E B} = \frac{C E}{E D}

\frac{6}{9} = \frac{8}{E D}

, giving

6 E D = 72

E D = 12

Questions students ask

How do you know which sides correspond in similar triangles?

By the order of the letters. In triangle ABC ~ triangle DEF, A matches D, B matches E and C matches F, so AB corresponds to DE.

Does a line parallel to one side make similar triangles?

Yes. A segment parallel to one side of a triangle cuts off a smaller triangle similar to the whole one.

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