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Similar triangles

# Similar triangles and indirect measurement

• The angles of similar triangles are equal.
• The sides of similar triangles are proportional.

Find u.

The original diagram included a smaller triangle and a larger triangle. Redraw them as separate triangles with corresponding sides in the same color.

ONM is similar to OLK because all three pairs of corresponding angles are congruent.

One pair of corresponding side lengths is MO and KO. Together they form the fraction $\frac{MO}{KO}$

Another pair of corresponding side lengths is ON and OL. Together they form the fraction $\frac{ON}{OL}$

In both fractions, the numerator comes from ONM and the denominator comes from OLK.

Since the triangles are similar, you can use the fractions to set up a proportion and solve for u.

 $\frac{MO}{KO}$
=
 $\frac{ON}{OL}$

 $\frac{5}{10}$
=
 $\frac{3}{u}$
Plug in the side lengths

5
 u
=
 3 × 10
Find the cross products

5
 u
= 30 Simplify

5
 u
÷ 5
=
 30 ÷ 5
Divide both sides by 5

 u
= 6

The missing length is 6 meters.

In the diagram below, DHG$\sim$ FHE. Find n.

 13 km 40 km 8 km

n = km