High School Geometry practice questions and exam guide

Two triangles are similar by AA (angle-angle). What can we conclude about corresponding sides?

1. Proportional (ratio of corresponding sides is constant — the scale factor)
2. Congruent
3. Equal lengths
4. Perpendicular
5. Parallel

Answer: A — Proportional (ratio of corresponding sides is constant — the scale factor)

Similar triangles: corresponding angles congruent, corresponding sides proportional. Scale factor = ratio of any pair of corresponding sides. AA, SAS, SSS are similarity criteria. Distinct from congruent (same size).

In a 45-45-90 triangle, if the legs equal 1, the hypotenuse equals:

1. √2
2. 2
3. 1
4. √3

Answer: A — √2

A) Standard ratio 1:1:√2. B/C/D) Each is incorrect.

A rigid motion (translation, rotation, or reflection):

1. Preserves distance and angle measure
2. Preserves distance but not angle measure
3. Preserves area but distorts distances
4. Changes all measurements proportionally

Answer: A — Preserves distance and angle measure

A) Rigid motions (isometries) preserve BOTH distance and angle measure — that's the definition. B) False — angles are preserved. C) False — area is preserved precisely because distances are. D) Describes a dilation (similarity), not a rigid motion.

If two triangles are similar with scale factor k, the ratio of their areas is:

1. k
2. k²
3. k³
4. 1

Answer: B — k²

B) Areas scale by k². A) Lengths. C) Volumes. D) Wrong.

A tangent line touches a circle at point P. The angle between the tangent and the radius to P is:

1. 0°
2. 45°
3. 60°
4. 90°

Answer: D — 90°

A) Treats them as the same line — false. B/C) Arbitrary guesses. D) Tangent-Radius Theorem: a tangent line is PERPENDICULAR to the radius drawn to the point of tangency.

Two similar solids have surface areas in ratio 9 : 25. The ratio of their volumes is:

1. 9 : 25
2. 3 : 5
3. 27 : 125
4. 81 : 625

Answer: C — 27 : 125

A) That's the surface-area ratio, not volume. B) Linear scale factor = √(9/25) = 3/5. C) Volumes scale by the CUBE of the linear factor: (3/5)³ = 27/125. D) Squared the surface-area ratio.

Surface area of a cube with edge s is:

1. s²
2. 4s²
3. 6s² — six faces of area s² each
4. s³

Answer: C — 6s² — six faces of area s² each

C) Standard formula. A/B/D) Each is incorrect.

Two triangles have side ratios 2:5. If smaller has area 12, larger has area:

1. 30
2. 75 (area scales as square of linear scale; (5/2)² = 25/4; 12 × 25/4 = 75)
3. 60
4. 100
5. 25

Answer: B — 75 (area scales as square of linear scale; (5/2)² = 25/4; 12 × 25/4 = 75)

Similar figures: area ratio = (linear ratio)². If linear ratio 2:5, area ratio 4:25. Larger area = 12 · (25/4) = 75. Volume scales as cube of linear ratio.

Two triangles are SIMILAR if:

1. Different shapes
2. Same orientation only
3. Same color
4. Same shape, proportional sides (corresponding angles equal, corresponding sides in same ratio)
5. Same size only

Answer: D — Same shape, proportional sides (corresponding angles equal, corresponding sides in same ratio)

Similar: same shape, sides proportional. Proof: AA (two angles equal), SAS similarity (one angle equal + sides around proportional), SSS similarity (all sides proportional). Scale factor preserves ratios.

The midpoint of (2, 4) and (8, 10) is:

1. (4, 5)
2. (5, 7) — midpoint = ((2+8)/2, (4+10)/2) = (5, 7)
3. (5, 5)
4. (10, 14)
5. (4, 7)

Answer: B — (5, 7) — midpoint = ((2+8)/2, (4+10)/2) = (5, 7)

Midpoint formula: M = ((x₁+x₂)/2, (y₁+y₂)/2). Average of x's, average of y's. (5, 7). Used for bisecting segments. Coordinate geometry basics.

Tangent to a circle is _____ to the radius at the point of tangency.

1. Parallel
2. Equal in length
3. Perpendicular (tangent ⊥ radius at point of tangency)
4. Bisected by the point of tangency (a common misconception — the radius and tangent line are perpendicular, not one bisecting the other)
5. Unrelated to the radius (in fact the two are directly connected: tangent ⊥ radius)

Answer: C — Perpendicular (tangent ⊥ radius at point of tangency)

Key circle theorem: tangent meets radius at 90°. Used in many proofs and right-triangle problems involving circles. G-C standards.

A cone has radius 3 and height 4. Its volume is:

1. 12π
2. 36π
3. 48π
4. 9π

Answer: A — 12π

A) V = (1/3)πr²h = (1/3)π(9)(4) = 12π. B) Forgot the 1/3 — that's the cylinder volume. C) Multiplied wrong factor. D) Used just r² without h.

In a right triangle with hypotenuse 13 and leg 5, find the other leg.

12. (169 − 25 = 144; √144 = 12.)

What is sin(45°)?

√2/2 ≈ 0.707.

Sum of exterior angles of any polygon (one at each vertex)?

Always 360°.

Reflect (4, 7) across the x-axis. Result?

(4, −7).

What is the 'inverse sine' (sin⁻¹) used for?

Finding the angle when you know the ratio. Like sin⁻¹(0.5) = 30°.

What's the relationship between a tangent and the radius at the point of tangency?

Perpendicular (90°).