Chapter 3 Review — Part 1

This first part gathers the key ideas of the chapter in one place: similar figures, proportion in triangles, transformations, circles, and the equation of a circle. Each block below is a compact recap with its main rules.

Similarity Similar polygons and triangles

Two polygons are similar when their corresponding angles are equal and their corresponding sides are in a constant ratio — the scale factor .

Triangles can be shown similar by three tests: AA (two equal angles), SSS (three sides in proportion), and SAS (two sides in proportion with the included angle equal).

Ratios that follow from : sides scale by , perimeters by , and areas by .

Proportion Proportion in triangles

Side-splitter: a line parallel to one side of a triangle cuts the other two sides in equal ratios. If , then .

Midsegment: the segment joining the midpoints of two sides is parallel to the third side and half its length: .

A neat application: the perpendicular bisector of any chord passes through the center of the circle.

Transformations Reflection, translation, rotation

Reflection over the x-axis:

Reflection over the y-axis:

Translation by

Rotation 90° about the origin:

Rotation 180° about the origin:

Two reflections over parallel lines make a translation; over intersecting lines they make a rotation. Every one of these transformations preserves lengths and angles, so the image stays congruent to the original.

Circle Angles, tangents, and lengths

Angles

Central angle = its intercepted arc.

Inscribed angle = half its arc; an inscribed angle on a diameter is 90°.

In a cyclic quadrilateral, opposite angles add to 180°.

Vertex inside the circle: angle = half the sum of the two arcs.

Vertex outside the circle: angle = half the difference of the two arcs.

Tangents and lengths

A tangent is perpendicular to the radius at the point of tangency.

Two tangents from an external point are equal in length.

Two chords crossing inside:

Two secants from outside:

Tangent and secant from outside:

Equation The equation of a circle

Center at the origin: . Center at : . The center's signs are reversed in the equation, and the right side is , not .

Example 1 — from center and radius

Center

, radius 5. Flip the signs and square the radius.

⟹ (x − 2)² + (y + 3)² = 25.

Example 2 — completing the square

Start from

Group and complete:

⟹ center (3, −2), radius 5.

Reference Quick review table

Topic	Key rule
Similar polygons	equal angles + sides in ratio k
Midsegment	parallel to the third side, half its length
Reflection over x	(x, y) → (x, −y)
Rotation 90°	(x, y) → (−y, x)
Inscribed angle	½ × its intercepted arc
Two chords inside	part × part = part × part
Equation of a circle	(x − a)² + (y − b)² = r²

Summary

Similarity: the scale factor links sides and perimeters, while links areas.
Proportion: a line parallel to a side makes equal ratios, and a midsegment is half the third side.
Transformations: reflection flips a sign, translation adds , and a 90° rotation swaps the coordinates with a sign change.
Circle: an inscribed angle is half the central one; inside a circle add the arcs, outside subtract them.
Equation of a circle: reverse the center's signs, and the right side is the radius squared.