In triangle , point lies on and point lies on such that . The ratio of to is to . If the area of triangle is , what is the area of trapezoid ?

Solution available with step-by-step explanation

SAT Lines, Angles & Triangles Question 33 | Free SAT Prep | Aniko

00:00

Question 33·Hard·Lines, Angles, and Triangles

0 / 50

In triangle $PQR$ , point $S$ lies on $PQ$ and point $T$ lies on $PR$ such that $ST \parallel QR$ . The ratio of $PS$ to $SQ$ is $3$ to $5$ . If the area of triangle $PST$ is $54$ , what is the area of trapezoid $STQR$ ?

For geometry questions with a segment parallel to a triangle’s side, immediately think similar triangles. First, turn any part-to-part ratio (like $PS:SQ$ ) into a part-to-whole ratio (such as $PS:PQ$ ). Use that as the linear scale factor between the small and large triangles, then square it to relate their areas. After finding the area of the entire triangle from the given smaller area, subtract to isolate the area of the remaining region (here, the trapezoid). This approach avoids needing any actual side lengths or heights and saves time on the test.

Hints

Click me!

Identify similar triangles

Because $ST$ is parallel to $QR$ , consider the relationship between triangle $PST$ and triangle $PQR$ . How does a line drawn parallel to one side of a triangle affect the angles?

Turn the given ratio into a whole-side ratio

You are given $PS:SQ = 3:5$ . What is $PQ$ in terms of these parts, and then what is the ratio $PS:PQ$ ?

Use area scaling for similar triangles

Once you know the ratio of a side in the small triangle to the corresponding side in the large triangle, how do their areas compare? Remember that area scales with the square of the side ratio.

Connect triangle areas to the trapezoid

After you find the area of the entire triangle $PQR$ , how can you use the area of $\triangle PST$ (which is given) to get the area of the trapezoid $STQR$ ?

Desmos Guide

Compute the area ratio between the small and large triangles

In Desmos, type (3/8)^2 to represent the area ratio $[\triangle PST]/[\triangle PQR] = (3/8)^2 = 9/64$ . Note the decimal or fraction value Desmos shows.

Find the area of the large triangle $PQR$

In a new expression line, type 54 / ((3/8)^2) to compute the area of $\triangle PQR$ . The output is the area of the big triangle.

Subtract the small triangle to get the trapezoid area

In another line, type 54 / ((3/8)^2) - 54. The value Desmos shows for this expression is the area of trapezoid STQR.

Step-by-step Explanation

Use the side ratio on segment $PQ$

We are told $PS:SQ = 3:5$ .

That means we can write:

$PS = 3k$
$SQ = 5k$
$PQ = PS + SQ = 3k + 5k = 8k$

So the ratio of the whole side $PQ$ to the part $PS$ is

\frac{PS}{PQ} = \frac{3k}{8k} = \frac{3}{8}.

Relate the two triangles using similarity

Since $ST \parallel QR$ , angle-angle (AA) similarity tells us that $\triangle PST \sim \triangle PQR$ .

In similar triangles, the ratio of corresponding sides is constant. Here,

\frac{PS}{PQ} = \frac{3}{8}

is the linear scale factor from $\triangle PQR$ (big) to $\triangle PST$ (small).

For areas, the scale factor is squared:

\frac{[\triangle PST]}{[\triangle PQR]} = \left(\frac{3}{8}\right)^2 = \frac{9}{64}.

Find the area of the large triangle $PQR$

We know the area of the smaller triangle $PST$ is $54$ .

Using the area ratio

\frac{[\triangle PST]}{[\triangle PQR]} = \frac{9}{64},

we solve for $[\triangle PQR]$ :

[\triangle PQR] = \frac{[\triangle PST]}{9/64} = 54 \cdot \frac{64}{9}.

Compute:

$54 \div 9 = 6$
$6 \cdot 64 = 384$

So the area of $\triangle PQR$ is $384$ .

Subtract to get the area of trapezoid $STQR$

The trapezoid $STQR$ is the region of the large triangle $PQR$ excluding the small triangle $PST$ .

So its area is

[STQR] = [\triangle PQR] - [\triangle PST] = 384 - 54 = 330.

Therefore, the area of trapezoid $STQR$ is 330, which corresponds to choice C.

Strategy

Hints

Desmos Guide

Step-by-step Explanation

Study Hints

Strategy

Hints

Desmos Guide

Step-by-step Explanation