In the standard (x, y) coordinate plane, a line segment has endpoints at P(2, 1) and Q(8, … — ACT

In the standard $(x, y)$ coordinate plane, a line segment has endpoints at $P(2, 1)$ and $Q(8, 5)$ . Line $L$ is parallel to segment $PQ$ and passes through the point $(3, -2)$ . If the point $(9, y)$ also lies on line $L$ , what is the value of $y$ ?

Answer: 2

Answer

The value of

y

2

The slope of segment

PQ

\frac{5 - 1}{8 - 2} = \frac{2}{3}

. Since line

L

is parallel to segment

PQ

, its slope is also

\frac{2}{3}

. The slope of line

L

through

(3, -2)

and

(9, y)

is given by

\frac{y - (-2)}{9 - 3} = \frac{y + 2}{6}

. Equating the two slopes yields

\frac{y + 2}{6} = \frac{2}{3}

, which simplifies to

y + 2 = 4

, so

y = 2

Step-by-Step Solution

Calculate the slope of segment

PQ

Slope of

PQ = \frac{2}{3}

Since parallel lines have equal slopes, finding the slope of the reference segment

PQ

is the first step in determining the slope of line

L

Set up the slope equation for line

L

using its parallel relationship to segment

PQ

\frac{y - (-2)}{9 - 3} = \frac{2}{3}

Because line

L

is parallel to segment

PQ

, its slope must also be

\frac{2}{3}

. The slope of line

L

is calculated using the points

(3, -2)

and

(9, y)

Solve the equation for

y

y = 2

Simplifying the numerator gives

y + 2

and the denominator gives

6

. Multiplying both sides of

\frac{y + 2}{6} = \frac{2}{3}

6

yields

y + 2 = 4

, which solves to

y = 2

Key Concept

Parallel lines have equal slopes in the coordinate plane.

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