Virginia SOL Math / Geometry / Triangles

G.TR.3

Triangle Similarity

What you'll learn

  • Prove triangles similar using AA, SSS, and SAS similarity
  • Find and use the scale factor between similar triangles
  • Solve for unknown lengths with proportions

Words to know

Similar —
same shape, sides in proportion, equal angles
Scale factor —
the ratio of matching side lengths
Proportion —
an equation stating two ratios are equal
AA similarity —
two pairs of equal angles prove similarity

Why it matters Similarity lets us find heights and distances we cannot reach, using shadows, models, and maps drawn to scale.

Worked examples

Model ItI do

The two triangles are similar. Find the scale factor from the small triangle to the large one.

Compare matching sides. The base grows from 3 to 6 and the height from 2 to 4. Each large side is twice the small side, so the scale factor is 6/3 = 2.

Try It TogetherWe do

Two similar triangles have matching sides 4 and 10. If another side of the small triangle is 6, find the matching side of the large triangle.

The scale factor is 10/4 = 2.5. Multiply: 6 x 2.5 = 15. The matching side is 15.

Now YouYou do

A 5 ft person casts a 4 ft shadow while a tree casts a 24 ft shadow at the same time. How tall is the tree?

Set up the proportion height/shadow = height/shadow. Check your answer in the back of this chapter.

Practice

Mastered 0 of 11

Warm-up

Ease in with the basics.

  1. 1.Which rule proves triangles similar from two pairs of equal angles?

  2. 2.Two similar triangles have matching sides 4 and 12. What is the scale factor?

  3. 3.The scale factor from a small triangle to a large one is 4. A small side is 6. Find the matching large side.

On Level

The core Quick Check from the book.

  1. 1.Which rule proves similarity from two pairs of equal angles?

  2. 2.Similar triangles have matching sides 3 and 12. What is the scale factor?

  3. 3.If the scale factor is 3 and a small side is 5, find the large side.

  4. 4.Are all equilateral triangles similar to each other?

  5. 5.Two similar triangles have sides 6 and 9. Find the matching side to a side of 8.

Challenge

Stretch your thinking.

  1. 1.A 6 ft person casts a 4 ft shadow while a flagpole casts a 30 ft shadow at the same time. How tall is the flagpole?

  2. 2.Two similar triangles have matching sides 8 and 20. A second side of the large triangle is 35. Find the matching side of the small triangle.

  3. 3.Triangle ABC is similar to triangle DEF with a scale factor of 3. If the perimeter of ABC is 15, find the perimeter of DEF.