Line A passes through the points (-3, 4) and (5, 8) in a coordinate plane. Line B is perpe… — ACT

Line $A$ passes through the points $(-3, 4)$ and $(5, 8)$ in a coordinate plane. Line $B$ is perpendicular to Line $A$ and passes through the point $(2, -1)$ . What is the $y$ -coordinate of the point where Line $B$ crosses the $y$ -axis?

Answer: 3

Answer

The correct answer is 3.

The correct answer is 3. Finding the slope of the first line gives 0.5. The negative reciprocal of 0.5 is -2, which is the slope of the perpendicular line. Setting up the slope equation between the point (2, -1) and the y-intercept (0, 3) confirms the y-coordinate is indeed 3.

Step-by-Step Solution

Calculate the slope of Line A.

The slope of Line A is 0.5.

Using the slope formula

m = \frac{y_2 - y_1}{x_2 - x_1}

for points

(-3, 4)

and

(5, 8)

gives

m_A = \frac{8 - 4}{5 - (-3)} = \frac{4}{8} = 0.5

Find the slope of Line B.

The slope of Line B is -2.

Because Line B is perpendicular to Line A, its slope must be the negative reciprocal of the slope of Line A:

m_B = -\frac{1}{0.5} = -2

Determine the y-intercept of Line B.

The y-coordinate of the y-intercept is 3.

Line B passes through

(2, -1)

and

(0, b)

. Using the slope formula:

\frac{b - (-1)}{0 - 2} = -2 \implies \frac{b + 1}{-2} = -2 \implies b + 1 = 4 \implies b = 3

Key Concept

The slope of a line measures its steepness, calculated as the change in

y

divided by the change in

x

. Perpendicular lines have slopes that are negative reciprocals of each other.

Alternative Method

Instead of using the slope formula directly to solve for the y-intercept, write the equation of Line B in point-slope form:

y - y_1 = m(x - x_1)

. Substituting the point

(2, -1)

and slope

-2

gives

y - (-1) = -2(x - 2)

, which simplifies to

y + 1 = -2x + 4

, or

y = -2x + 3

. To find the y-intercept, set

x = 0

, which yields

y = 3

Estimated Time:1m 30s

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Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

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