In the standard (x, y) coordinate plane, a line segment has midpoint M(4, 1) and one endpo… — ACT

In the standard $(x, y)$ coordinate plane, a line segment has midpoint $M(4, 1)$ and one endpoint $P(x, 5)$ . If the total length of the line segment is $10$ units, and $x < 4$ , what is the value of $x$ ?

Answer: 1

Answer

The midpoint

M

divides the line segment into two equal segments, meaning the distance from endpoint

P

to midpoint

M

is half of the total length of the segment,

\frac{10}{2} = 5

units. Applying the distance formula between

P(x, 5)

and

M(4, 1)

gives

\sqrt{(x - 4)^2 + (5 - 1)^2} = 5

. Squaring both sides yields

(x - 4)^2 + 16 = 25

, which simplifies to

(x - 4)^2 = 9

. Solving for

x

gives

x - 4 = 3

x - 4 = -3

, meaning

x = 7

x = 1

. The problem specifies that

x < 4

, so the only valid value is

1

Step-by-Step Solution

Find the distance from the endpoint to the midpoint.

The distance

PM

5

units.

The midpoint divides the line segment into two equal parts, so the distance from any endpoint to the midpoint is half of the total length of the segment:

\frac{10}{2} = 5

Set up the distance formula equation for the segment

PM

\sqrt{(x - 4)^2 + (5 - 1)^2} = 5

Using the coordinates of

P(x, 5)

and

M(4, 1)

with the distance formula

d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

Simplify the equation and square both sides.

(x - 4)^2 + 16 = 25

Calculating

(5 - 1)^2 = 16

and squaring both sides of the equation to eliminate the radical.

Isolate the squared term and solve for

x

x = 7

x = 1

Subtracting

16

from both sides gives

(x - 4)^2 = 9

. Taking the square root gives

x - 4 = 3

x - 4 = -3

, yielding

x = 7

x = 1

Apply the constraint

x < 4

x = 1

Since the question states that

x < 4

, we choose

x = 1

instead of

x = 7

Key Concept

The midpoint divides a segment into two segments of equal length, and the distance between two points

(x_1, y_1)

and

(x_2, y_2)

is given by

\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

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