Find x, y, and AI in the parallelogram

Question

In parallelogram KASI, the diagonals AI and KS intersect at P. If KP=2x, PS= y-1, AP= 2y-1, and IP= 2x+3, find x, y, and AI.

Answer 1

Diagonals of a parallelogram bisect one another so KP = PS and AP = IP

So you have two equations and two unknowns:

2x = y - 1

2y - 1 = 2x + 3

put into standard form

2x - y = -1

2x - 2y = -4

subtract second equation from first;

y = 3

plug in value of y into 1st equation and solve for x:

2x - (3) = -1

2x = 2

x = 1

Plug in values of x and y to compute AP and IP:

AP = 2y - 1 = 2(3) - 1 = 5

IP = 2x + 3 = 2(1) + 3 = 5

AI = AP + IP = 10