Introduction to the Problem

The problem at hand involves determining the coordinates of a vertex, R, in a triangle PQR given specific conditions. Specifically, points P and Q are given, the vertex R lies on a particular line, and the distance PR is half the distance PQ. This problem requires the application of geometric principles and the distance formula to find the coordinates of R.

Understanding the Given Information

We begin with the coordinates of the points P and Q, where P is at (-3, 4) and Q is at (3, -2). The vertex R is constrained to lie on the line y 1. Additionally, the distance between P and R is half the distance between P and Q.

Step 1: Calculating the Distance PQ

To find the length of PQ, the distance formula is used:

Distance formula: d sqrt{(x_2 - x_1)^2 (y_2 - y_1)^2}

For points P(-3, 4) and Q(3, -2):

PQ sqrt{(3 - (-3))^2 (-2 - 4)^2} sqrt{6^2 (-6)^2} sqrt{36 36} sqrt{72} 6sqrt{2}

Step 2: Determining the Length of PR

According to the problem, PR is half the length of PQ:

PR (1/2) * PQ (1/2) * 6sqrt{2} 3sqrt{2}

Step 3: Expressing R on the Line y 1

Since R lies on the line y 1, we can represent the coordinates of R as (x, 1).

Step 4: Applying the Distance Formula for PR

Using the distance formula for PR, we set up the equation:

sqrt{(x - (-3))^2 (1 - 4)^2} 3sqrt{2}

This simplifies to:

sqrt{(x 3)^2 (-3)^2} 3sqrt{2}

Squaring both sides gives:

(x 3)^2 9 18

(x 3)^2 9

x 3 ±3

x 0 or x -6

Step 5: Determining the Coordinates of R

Thus, the two possible coordinates for R are:

- When x 0: R(0, 1)

- When x -6: R(-6, 1)

Conclusion and Verification

The problem has been solved, and the coordinates of R are determined as (-6, 1) and (0, 1). We verify the solution by checking that the distance PR is indeed half of PQ for both coordinates.

Therefore, the two possible positions of R are R(-6, 1) and R(0, 1).