college geometry

Question

In the diagram below, LM⎯⎯⎯⎯⎯⎯⎯⎯∥NO⎯⎯⎯⎯⎯⎯⎯⎯⎯,  LN=x−1,  KM=x+2,  MO=2,  and  KL=20.  (Note that the diagram may not be drawn to scale.)  Find KO.

a) KO=6

b) KO=8

c) KO=10

d) KO=11

e) KO=7

f) None of the above

Answer 1

The diagram above two Similar Triangles with proportional sides are formed.

The sides of the inner Smaller Top Triangle LKM are KL = 20 and KM = x + 2

The sides of the outer Larger Triangle NKO are KN = KL + LN = 20 + x - 1 and

KO = KM + MO = x + 2 + 2

If you ratio sides to one another, and solve for x then you can plug that value in to find KO.

(x + 2) +2 = ( x +2)

20 + x - 1 (20)

Combine like terms where applicable then cross multiply to get the products belo

20( x + 4) = (19 + x)(x + 2)

20x + 80 = x² + 19x + 2x + 38

20x + 80 = x² + 21x + 38

Subtract the quantity (20x + 80) from both sides to apply the quadratic below

0 = x² +21x -20x - 80 + 38

0= x² + x - 42

You can factor this and use the positive root since x cannot be negative.

You can check it by substituting in your value for x to get the sides for small and large triangles and checking the ratio.

I got C. KO =10

Give it a try, I hope your find this useful and send me a message if you have any questions.