Exercise 4 Page 577

We are given a triangle △JKL and we want to determine whether the segments JL and MP are parallel.

By the Converse Triangle Proportionality Theorem, if a line intersects two sides of a triangle and separates the sides into proportional corresponding segments, then the line is parallel to the third side of the triangle. Let's check if the sides are divided proportionally. JM/MK ? = LP/PKSince we are given the lengths of JM and JK, we can find the length of MK.

JM+MK=JK

SubstituteII

JM= 5, JK= 15

5+MK= 15

SubEqn

LHS-5=RHS-5

MK=10

Similarly we are given the lengths of PK and LK, so we can find the length of LP.

LP+PK=LK

SubstituteII

PK= 9, LK= 13

LP+ 9= 13

SubEqn

LHS-9=RHS-9

LP=4

Now we can substitute all known lengths into our proportion. JM/MK ? = LP/PK ⇕ 5/10 ? = 4/9 We will simplify each ratio as much as possible to see if they are equivalent.

5/10 ? = 4/9

ReduceFrac

a/b=.a /5./.b /5.

1/2 ≠ 4/9

As we can see, the sides are not divided proportionally. Therefore, segments JL and MP are not parallel.