SAT Lines, Angles & Triangles Question 9 | Free SAT Prep

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Question 9·Hard·Lines, Angles, and Triangles

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Note: Figure not drawn to scale.

In the figure shown, $\overline{RS}$ and $\overline{MN}$ intersect at point $T$ . The angles at $M$ and $S$ are congruent (each marked $b^{\circ}$ ). $TM = 35$ , $TS = 25$ , $MR = 56$ , and $TN = 60$ . What is the length of $\overline{SN}$ ?

Your answer

For geometry questions with intersecting lines and several given side lengths, first scan the diagram for marked equal angles and vertical angles at intersections; these usually signal similar triangles. Identify which triangles share those angles, then carefully match corresponding vertices (often via the equal angles) to pair up sides correctly. Set up a proportion using sides that are both known and adjacent to the equal angles, solve algebraically for the unknown side, and ignore extra lengths that do not fit into the similarity relationship.

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Look for triangles that share the marked angles

Which two triangles in the figure each contain one of the angles marked $b^\circ$ at $M$ and $S$ , and also involve the given side lengths (35, 25, 56, and the unknown)? Focus on triangles that have vertices at $M$ , $T$ , $R$ and at $S$ , $T$ , $N$ .

Use the intersection at T

At point $T$ , lines $RS$ and $MN$ intersect and form vertical angles. Which angles in triangles $MTR$ and $STN$ are vertical to each other, and what does that tell you about the relationship between the two triangles?

Set up a ratio of corresponding sides

Once you know the two triangles are similar, match the angle at $M$ with the angle at $S$ . Then compare the sides that touch those angles: use the sides of lengths 35 and 25 in one ratio, and match them with the sides of lengths 56 and the unknown length in another ratio.

Solve the equation you get

Your proportion should look like a fraction with 35 and 25 on one side and 56 and the unknown on the other. Solve this proportion for the unknown side length.

Desmos Guide

Write the equation for SN from similarity

From the similar triangles, you should have a proportion of the form $35/25 = 56/x$ , where $x$ represents the length of $SN$ . Rearrange it to get $x = (25\cdot 56)/35$ .

Evaluate the expression in Desmos

In Desmos, type (25*56)/35 (or 56*25/35) and let Desmos compute the result. The value it outputs is the length of $SN$ .

Step-by-step Explanation

Identify the key triangles

Focus on triangles $MTR$ and $STN$ :

Triangle $MTR$ includes the sides $TM=35$ and $MR=56$ , and has an angle at $M$ marked $b^\circ$ .
Triangle $STN$ includes the sides $TS=25$ and $SN$ (the unknown), and has an angle at $S$ also marked $b^\circ$ .

These are the two non-right triangles that involve all the given side lengths and the marked angles.

Use the angle information to show similarity

Because the angles at $M$ and at $S$ are both marked $b^\circ$ , we know

$\angle M$ in triangle $MTR$ equals $\angle S$ in triangle $STN$ .

Also, lines $RS$ and $MN$ intersect at $T$ , so the angle $\angle MTR$ in triangle $MTR$ and the angle $\angle STN$ in triangle $STN$ are vertical angles, which are congruent.

With two pairs of equal angles, triangles $MTR$ and $STN$ are similar by the angle-angle (AA) criterion.

Match corresponding vertices and set up a proportion

From the similarity, we match vertices by their equal angles:

$M$ in triangle $MTR$ corresponds to $S$ in triangle $STN$ (both have angle $b^\circ$ ).
$T$ corresponds to $T$ (they share the vertical angle at $T$ ).
$R$ corresponds to $N$ (the remaining vertices).

So the corresponding sides are:

$TM$ ↔ $TS$
$MR$ ↔ $SN$
$TR$ ↔ $TN$

Use the sides around the $b^\circ$ angles to make a proportion:

\frac{TM}{TS} = \frac{MR}{SN}

Substitute the known values:

\frac{35}{25} = \frac{56}{SN}

Solve the proportion for SN

Solve

\frac{35}{25} = \frac{56}{SN}

by cross-multiplying:

35\cdot SN = 25\cdot 56

Now divide both sides by $35$ :

SN = \frac{25\cdot 56}{35}

Simplify the fraction:

$25/35 = 5/7$ , so

SN = \frac{5\cdot 56}{7} = 5\cdot 8 = 40

So the length of $\overline{SN}$ is $40$ .

Strategy

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Desmos Guide

Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation