college geometry

Question

In the diagram below, KL⎯⎯⎯⎯⎯∥HP⎯⎯⎯⎯⎯⎯∥GJ⎯⎯⎯⎯⎯.  PJ is 5 times greater than KH , LP is 8 units greater than KH , and HG is 4 units smaller than PJ . Find the value of KH .

(Note that the diagram may not be drawn to scale.)

a) KH=32

b) KH=45

c) KH=811

d) KH=8

e) KH=89

f) None of the above

Answer 1

O is the intersection of transversal stray lines GHK and JPL.

Corresponding angles are congruent per the given parallel lines.

Dropping the perpendicular from K to HP and GJ produces

similar triangles, since the corresponding angles are congruent.

LIkewise, dropping the perpendicular from L to HP and GJ produces

another pair of similar triangles.

Specifically, HK is proportional to HG just as LP is proportional to PJ

From the given, HK = x, HG = 5x-4, LP = x+8, and PJ = 5x

Then

x/ (6x-4) = (x+8)/(6x+8)

cross multiplciation :

x(6x+8) = (x+8)(6x-4)

6x^2 + 8x = 6x^2 + 44x - 32

8x = 44x - 32

0 = 36x - 32

32/26 = x

8/9 = x = HK

None of the answers work in this scenario , while x=8/9 produces a 2 to 1 ratio between

these related sides