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 $k$ .

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 $k$ : sides scale by $k$ , perimeters by $k$ , and areas by $k^2$ .

Proportion Proportion in triangles

Side-splitter: a line parallel to one side of a triangle cuts the other two sides in equal ratios. If $DE \parallel BC$ , then $\dfrac{AD}{DB} = \dfrac{AE}{EC}$ .

Midsegment: the segment joining the midpoints of two sides is parallel to the third side and half its length: $MN = \tfrac{1}{2}\,BC$ .

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:

(x, y) \to (x,\, -y)

Reflection over the y-axis:

(x, y) \to (-x,\, y)

Translation by

(a, b)

(x, y) \to (x+a,\; y+b)

Rotation 90° about the origin:

(x, y) \to (-y,\, x)

Rotation 180° about the origin:

(x, y) \to (-x,\, -y)

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:

a \times b = c \times d

Two secants from outside:

e_1 \times s_1 = e_2 \times s_2

Tangent and secant from outside:

t^{2} = e \times s

Equation The equation of a circle

Center at the origin: $x^{2} + y^{2} = r^{2}$ . Center at $(a, b)$ : $(x-a)^{2} + (y-b)^{2} = r^{2}$ . The center's signs are reversed in the equation, and the right side is $r^{2}$ , not $r$ .

Example 1 — from center and radius

Center

(2, -3)

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

⟹ (x − 2)² + (y + 3)² = 25.

Example 2 — completing the square

Start from

x^{2} + y^{2} - 6x + 4y - 12 = 0

Group and complete:

(x-3)^{2} - 9 + (y+2)^{2} - 4 = 12

(x-3)^{2} + (y+2)^{2} = 25

⟹ 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 $k$ links sides and perimeters, while $k^2$ 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 $(a, b)$ , 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.