In the standard (x, y) coordinate plane, a triangle has vertices at D(1, -2), E(4, 7), and… — ACT

In the standard $(x, y)$ coordinate plane, a triangle has vertices at $D(1, -2)$ , $E(4, 7)$ , and $F(6, k)$ . If the line containing the altitude from vertex $F$ to side $DE$ has a $y$ -intercept of $8$ , what is the value of $k$ ?

A
-10
B
$\frac{2}{3}$
6Answer
D
10
E
26

Answer

The value of

k

6

The slope of side

DE

is calculated as

\frac{7 - (-2)}{4 - 1} = 3

. An altitude is perpendicular to the side it intersects, so the slope of the altitude is the negative reciprocal of

3

, which is

-\frac{1}{3}

. The line containing this altitude has a

y

-intercept of

8

, giving the equation

y = -\frac{1}{3}x + 8

. Since the vertex

F(6, k)

lies on this line, substituting

x = 6

and

y = k

into the equation yields

k = -\frac{1}{3}(6) + 8 = -2 + 8 = 6

. This matches the correct value of

6

Step-by-Step Solution

Calculate the slope of side

DE

using the coordinates of

D(1, -2)

and

E(4, 7)

The slope of

DE

3

The slope formula is

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

. Substituting the points gives

m_{DE} = \frac{7 - (-2)}{4 - 1} = \frac{9}{3} = 3

Determine the slope of the altitude line, which is perpendicular to side

DE

The slope of the altitude is

-\frac{1}{3}

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

3

-\frac{1}{3}

Write the equation of the line containing the altitude using its slope and the given

y

-intercept of

8

The equation of the altitude line is

y = -\frac{1}{3}x + 8

The slope-intercept form of a linear equation is

y = mx + b

, where

m

is the slope and

b

is the

y

-intercept.

Substitute the coordinates of point

F(6, k)

into the equation of the altitude to solve for

k

k = 6

Since vertex

F

lies on the line containing the altitude, its coordinates must satisfy the equation:

k = -\frac{1}{3}(6) + 8 = -2 + 8 = 6

Key Concept

Finding the equation of a line perpendicular to a given line segment and using it to find a missing coordinate of a point on that line.

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