In triangle PQR, a line segment parallel to QR intersects sides PQ and PR at S and T, resp… — SAT

In triangle $PQR$ , a line segment parallel to $QR$ intersects sides $PQ$ and $PR$ at $S$ and $T$ , respectively. If the length of segment $PS$ is $2x - 1$ , the length of segment $SQ$ is $3$ , the length of segment $PT$ is $x + 2$ , and the length of segment $TR$ is $2$ , what is the length of segment $PS$ ?

A
2.5
B
8
15Answer
D
13

Answer

The length of segment

PS

is 15.

According to the Triangle Proportionality Theorem, if a line is parallel to one side of a triangle and intersects the other two sides, then it divides the two sides proportionally. Therefore, we can write the proportion as

\frac{PS}{SQ} = \frac{PT}{TR}

. Substituting the given values yields

\frac{2x - 1}{3} = \frac{x + 2}{2}

. Cross-multiplying gives

2(2x - 1) = 3(x + 2)

, which simplifies to

4x - 2 = 3x + 6

. Subtracting

3x

and adding

2

to both sides results in

x = 8

. Substituting

x = 8

back into the expression for

PS

gives

2(8) - 1 = 15

. Thus, the length of segment

PS

is 15.

Step-by-Step Solution

Set up a proportion using the Triangle Proportionality Theorem.

\frac{PS}{SQ} = \frac{PT}{TR}

Since segment

ST

is parallel to side

QR

, it divides the sides of triangle

PQR

proportionally.

Substitute the given algebraic expressions into the proportion.

\frac{2x - 1}{3} = \frac{x + 2}{2}

This establishes a solvable equation for

x

based on the geometric relationship.

Cross-multiply and solve for

x

2(2x - 1) = 3(x + 2) \implies 4x - 2 = 3x + 6 \implies x = 8

Cross-multiplication removes the denominators, allowing us to isolate the variable

x

Calculate the length of segment

PS

using the value of

x

PS = 2(8) - 1 = 15

The question asks for the length of segment

PS

, so we must evaluate the expression

2x - 1

x = 8

Key Concept

Triangle Proportionality Theorem and Similarity

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

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