In the standard (x, y) coordinate plane, the points A(1, 2) and B(9, 8) are the endpoints … — ACT

In the standard $(x, y)$ coordinate plane, the points $A(1, 2)$ and $B(9, 8)$ are the endpoints of a diameter of a circle $C$ . A line $L$ passes through the center of $C$ and is perpendicular to segment $AB$ . A point $P(x, y)$ lies on line $L$ such that the distance from $P$ to the center of $C$ is equal to the radius of $C$ . If the $x$ -coordinate of $P$ is greater than the $x$ -coordinate of the center of $C$ , what is the $y$ -coordinate of $P$ ?

Answer: 1

Answer

The y-coordinate of the point P is 1.

The correct answer is 1. The center of circle C is the midpoint of the diameter AB, which is calculated as M(5, 5). The radius is half the length of AB, which is 5. The line L passing through M perpendicular to AB has a slope of -4/3. Points on this line at a distance of 5 from M are found by changing the coordinates by (+3, -4) or (-3, +4), yielding (8, 1) and (2, 9). Since the x-coordinate must be greater than the center's x-coordinate of 5, the correct point is (8, 1), which has a y-coordinate of 1.

Step-by-Step Solution

Calculate the center of the circle C by finding the midpoint of the diameter AB.

The center is M(5, 5).

The center of a circle is the midpoint of any of its diameters.

Calculate the radius of circle C by finding half the distance between A(1, 2) and B(9, 8).

The radius is 5.

The distance formula gives the diameter length as 10, and the radius is half the diameter.

Find the slope of line L perpendicular to AB.

The slope of L is -4/3.

The slope of AB is 3/4, and perpendicular lines have slopes that are negative reciprocals of each other.

Determine the coordinates of point P using the distance from the center and the slope of line L.

The possible points are (8, 1) and (2, 9).

Moving a distance of 5 along a line with slope -4/3 from (5, 5) results in a change of +/-3 in the x-coordinate and -/+4 in the y-coordinate.

Apply the constraint that the x-coordinate of P must be greater than the x-coordinate of the center (5).

P is (8, 1), so the y-coordinate is 1.

Comparing the two candidate points, only (8, 1) has an x-coordinate greater than 5.

Key Concept

Applying midpoint, distance, and perpendicular slope relationships in coordinate geometry to locate points.

Estimated Time:2m 30s

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