Question

strategy we will use the midpoint formula to find the unknown coordinates. we need to find ( x_2 ) and ( y_2 ).
why we want to find the coordinates of one of the endpoints.
solution we can let ( p(x_1, y_1) = p(-5, -3) ) and ( m(x_m, y_m) = m(-1, 2) ), where ( x_m ) represents the ( x )-coordinate of point ( m ), and ( y_m ) represents the ( y )-coordinate of point ( m ). we can then find the coordinates of point ( q ) using the midpoint formula.
( x_m = \frac{x_1 + x_2}{2} ) and ( y_m = \frac{y_1 + y_2}{2} )
( -1 = \frac{-5 + x_2}{2} ) ( 2 = \frac{-3 + y_2}{2} )
( square = -5 + x_2 ) ( 4 = square + y_2 ) multiply both sides by 2.
( 3 = x_2 ) ( 7 = y_2 )
since ( x_2 = 3 ) and ( y_2 = 7 ), the coordinates of point ( q ) are ( (x, y) = (square) )
self check if the midpoint of a segment joining ( pq ) is ( m(4, -9) ) and one endpoint is ( q(-4, 1) ), find the coordinates of point ( p ).
( (x, y) = (square) )

Explanation:

Step1: Solve for x₂ in x-coordinate formula

Given \(x_M = \frac{x_1 + x_2}{2}\), \(x_M = -1\), \(x_1 = -5\).
Multiply both sides by 2: \(-1 \times 2 = -5 + x_2\) → \(-2 = -5 + x_2\).
Add 5 to both sides: \(x_2 = 3\).

Step2: Solve for y₂ in y-coordinate formula

Given \(y_M = \frac{y_1 + y_2}{2}\), \(y_M = 2\), \(y_1 = -3\).
Multiply both sides by 2: \(2 \times 2 = -3 + y_2\) → \(4 = -3 + y_2\).
Add 3 to both sides: \(y_2 = 7\).

Step3: Find coordinates of Q

Point Q is \((x_2, y_2) = (3,7)\).

Step4: Solve Self Check for P's x-coordinate

Let \(P(x_1,y_1)\), \(Q(-4,1)\), midpoint \(M(4,-9)\).
Using \(x_M = \frac{x_1 + x_2}{2}\): \(4 = \frac{x_1 + (-4)}{2}\).
Multiply by 2: \(8 = x_1 - 4\) → \(x_1 = 12\).

Step5: Solve Self Check for P's y-coordinate

Using \(y_M = \frac{y_1 + y_2}{2}\): \(-9 = \frac{y_1 + 1}{2}\).
Multiply by 2: \(-18 = y_1 + 1\) → \(y_1 = -19\).

Answer:

(3, 7)
(12, -19)