In triangle PQR, point S lies on side PQ and point T lies on side PR such that segment ST … — SAT

In triangle $PQR$ , point $S$ lies on side $PQ$ and point $T$ lies on side $PR$ such that segment $ST$ is parallel to segment $QR$ . The length of segment $PS$ is $x$ , the length of segment $SQ$ is $6$ , the length of segment $ST$ is $x + 2$ , and the length of segment $QR$ is $2x + 7$ . What is the length of segment $QR$ ?

Answer: 15

Answer

The length of segment

QR

15

Since segment

ST

is parallel to segment

QR

, triangle

PST

is similar to triangle

PQR

. The ratio of their corresponding sides is equal, so

\frac{PS}{PQ} = \frac{ST}{QR}

. Substituting the values gives

\frac{x}{x + 6} = \frac{x + 2}{2x + 7}

. Cross-multiplying and simplifying yields

x^2 - x - 12 = 0

. Factoring gives

(x-4)(x+3)=0

, so

x=4

because side lengths must be positive. Substituting

x=4

into

2x+7

yields

15

Step-by-Step Solution

Determine the similarity between the triangles

Triangle

PST

is similar to triangle

PQR

Since segment

ST

is parallel to segment

QR

, the corresponding angles are equal, establishing similarity by Angle-Angle (AA) criterion.

Express the total length of side

PQ

PQ = x + 6

The length of side

PQ

is the sum of the collinear segments

PS

and

SQ

Set up the similarity ratio equation

\frac{x}{x + 6} = \frac{x + 2}{2x + 7}

Corresponding side lengths of similar triangles are in proportion:

\frac{PS}{PQ} = \frac{ST}{QR}

Solve the quadratic equation for

x

x = 4

Cross-multiplying gives

x(2x + 7) = (x + 2)(x + 6) \implies 2x^2 + 7x = x^2 + 8x + 12 \implies x^2 - x - 12 = 0 \implies (x - 4)(x + 3) = 0

. Since length must be positive,

x = 4

Calculate the length of segment

QR

QR = 15

Substitute

x = 4

into the expression for

QR

, which is

2x + 7

, resulting in

2(4) + 7 = 15

Key Concept

Using triangle similarity and algebraic equations to find unknown segment lengths.

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