DWG NO. 07.2 — Lesson 2 of 5

Similar Polygons

Unit 7: Similarity · ~15–30 min

Objective

Determine whether two polygons are similar, and use the scale factor to find missing side lengths and perimeters.

What "similar" means

Two polygons are similar (symbol ∼) when they have the exact same shape but not necessarily the same size. Formally, two polygons are similar when both conditions hold:

Corresponding angles are congruent. Every angle in one polygon matches an equal angle in the other.
Corresponding sides are proportional. The ratio of each pair of matching sides is the same constant — the scale factor.

As with congruence statements, the order of the letters in a similarity statement like ▵ABC ∼ ▵DEF tells you the correspondence: A matches D, B matches E, C matches F. Get the order right, or the ratios you set up will be wrong.

Scale factor and perimeter

The scale factor is the constant ratio between corresponding sides, usually written in simplest form (e.g. 2:3). A useful shortcut: the ratio of the polygons' perimeters equals the same scale factor — you don't need to add up every side to compare perimeters, just scale the whole number.

Square ABCD ∼ square EFGH: angles match (all 90°) and sides scale by the same factor.

Worked Example 1 · Finding a missing side

Problem▵ABC ∼ ▵DEF. AB = 6, DE = 9, and BC = 8. Find EF.

1.The scale factor from ▵ABC to ▵DEF is AB/DE = 6/9 = 2/3.

2.Set up the proportion for the other pair: BC/EF = 2/3, so 8/EF = 2/3, giving 2(EF) = 24, EF = 12.

EF = 12

Worked Example 2 · Perimeter from scale factor

ProblemPentagon JKLMN ∼ pentagon PQRST with scale factor 3:5. The perimeter of JKLMN is 24. Find the perimeter of PQRST.

1.Perimeters scale the same way as sides: 24/x = 3/5.

2.Cross multiply: 3x = 120, so x = 40.

Perimeter of PQRST = 40

Guided practice

1.▵RST ∼ ▵XYZ. If RS = 10 and XY = 15, what is the scale factor from ▵RST to ▵XYZ?
Show answer
10/15 = 2/3, so the scale factor is 2:3.
2.Using the scale factor 2:3 from problem 1, if ST = 8, find YZ.
Show answer
8/YZ = 2/3, so 2(YZ) = 24, YZ = 12.
3.Two similar rectangles have a scale factor of 4:7. The smaller rectangle has a perimeter of 32. Find the perimeter of the larger rectangle.
Show answer
32/x = 4/7, so 4x = 224, x = 56.
4.Quadrilateral WXYZ ∼ quadrilateral PQRS. Name the two angles that must be congruent to ∠X, and the side that corresponds to segment YZ.
Show answer
Only one angle corresponds to ∠X: ∠Q (matching letter position, second in each name). The side corresponding to YZ is RS.
5.Explain why two squares are always similar to each other, no matter their side lengths.
Show answer
Every angle in any square is 90°, so corresponding angles are automatically congruent. And since all four sides of a square are equal, the ratio of any side in one square to any side in the other is the same constant. Both similarity conditions are satisfied automatically.

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