In the diagram, $ABCD$ is a square. Find $PR.$

The side length of the square is 10. L, M, N, O are midpoints.

tomtom Dec 18, 2023

#1**+1 **

Let A = (0,0)

Let D = (0,10)

Let C= (10,10)

Let B = (10,0)

Line through DL has the equation y = (-2)x + 10

Line through AM has the equation y= (1/2)x

x intersection of these lines

-2x + 10 = 1/2x

-(5/2)x = -10

x = 4

y= 2

P = (4,2)

Line through OC has the equation y = (1/2)x + 5

Line through NV has the equation y = (-2)(x -10) = -2x + 20

x intersection of these lines

(-2x) + 20 = (-1/2)x + 5

(-5/2)x = -15

x = 6

y = 8

R = (6,8)

PR = sqrt [ (4 - 6)^2 + (2 - 8)^2 ] = sqrt [ 4 + 36 ] = sqrt 40 = 2sqrt (10)

CPhill Dec 19, 2023