Exercise 13 Page 820

First, let's consider the given diagram and add the information about the side lengths.

Since MN ∥ OP we can conclude that ∠ NML and ∠ POL, so ∠ MNL and ∠ OPL are corresponding angles. Therefore, by the Corresponding Angles Theorem we can tell that ∠ NML≅ ∠ POL and ∠ MNL≅∠ OPL.

Since two angles of △ MLN are congruent to two angles of △ OPL, these two triangles are similar by the Angle-Angle Similarity Postulate. Therefore, the lengths of the sides of the triangles are proportional. ML/OL=MN/OP Knowing that MN= 15, ML= 12, OM= 6, and the fact that we can express the length of OL as the sum of lengths of OM and ML, we can substitute these lengths into our proportion and solve it.

ML/OL=MN/OP

Substitute

OL= OM+ML

ML/OM+ML=MN/OP

SubstituteValues

Substitute values

12/6+ 12=15/OP

▼

Solve for OP

AddTerms

Add terms

12/18=15/OP

MultEqn

LHS * OP=RHS* OP

12/18 * OP=15

MultEqn

LHS * 18=RHS* 18

12* OP=270

DivEqn

.LHS /12.=.RHS /12.

OP=22.5