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Geometry Unit 4: Similarity & right triangles

Similar figures have the same shape but different size. Their sides are proportional, and right triangles have special relationships of their own.

Covers: Scale factor, AA similarity, proportions, Pythagorean theorem, special right triangles

Study guide

Similar figures have the same shape but not necessarily the same size: corresponding angles are equal and corresponding sides are proportional. The scale factor k is the ratio of corresponding lengths. Perimeters scale by k, areas by k², and volumes by k³.

Triangles can be proven similar with AA (two pairs of equal angles), SAS similarity (two sides in proportion with the included angle equal), or SSS similarity (all three sides in proportion). Once triangles are similar, set up a proportion of corresponding sides and cross-multiply to find missing lengths. Shadow and indirect-measurement problems are classic applications.

The Pythagorean theorem, a² + b² = c², works for right triangles, where c is the hypotenuse. Its converse classifies triangles: if a² + b² = c², the triangle is right; if a² + b² > c² it's acute; if a² + b² < c² it's obtuse. Pythagorean triples like 3-4-5, 5-12-13, 8-15-17 and 7-24-25 are worth recognizing.

Special right triangles have fixed ratios. In a 45-45-90 triangle the legs are equal and the hypotenuse is leg·√2. In a 30-60-90 triangle the sides are x (opposite 30°), x√3 (opposite 60°) and 2x (hypotenuse).

A segment joining the midpoints of two sides of a triangle is parallel to the third side and half as long. A line parallel to one side of a triangle that crosses the other two sides divides them proportionally. Conversely, if a line divides two sides proportionally, it is parallel to the third side, which is how proportional segments prove lines parallel.

The altitude to the hypotenuse of a right triangle forms two smaller triangles similar to the original and to each other. If the hypotenuse is split into segments p and q, then the altitude satisfies h² = p·q, and each leg satisfies leg² = (adjacent segment)·(whole hypotenuse). Match sides by corresponding angles, not by position on the page, and keep the same order in both ratios (small/large = small/large). Writing the similarity statement with vertices in matching order, such as △ABC ~ △DEF, tells you which sides correspond.

Key ideas

• AA: two equal angles make triangles similar.
• Corresponding sides are proportional with scale factor k.
• Perimeters scale by k; areas by k².
• Pythagorean theorem: a² + b² = c².
• Special triangles: 45-45-90 is x, x, x√2; 30-60-90 is x, x√3, 2x.

Formulas and rules

Common mistakes

• Setting up a proportion with sides that don't correspond.
• Putting the √3 on the wrong leg in a 30-60-90 triangle.

Words to know

AA similarity

Two pairs of equal angles make triangles similar.

Scale factor k (area)

Areas scale by k².

Pythagorean triples

3-4-5, 5-12-13, 8-15-17

30-60-90 sides

x (short), x√3 (long), 2x (hypotenuse)

Pythagorean triple

Whole numbers with a² + b² = c², like 3-4-5 or 7-24-25.

45-45-90 triangle

Sides x, x, x√2.

Similar figures

Same shape; equal angles and proportional sides.

Scale factor

The ratio of corresponding lengths in similar figures.

SAS similarity

Two sides in proportion with the included angle equal.

Pythagorean theorem

a² + b² = c²

Converse of the Pythagorean theorem

If a² + b² = c², the triangle is right; > means acute, < means obtuse.

Indirect measurement

Using similar triangles (like shadows) to find heights.

Sample practice questions

39 more questions like these, from level 1 to 5, are in the practice set.