Point P is at (4, −2). After reflecting over the x-axis, where is P'? — (4, 2).Reflecting over the x-axis: (x, y) → (x, −y). P'(4, −(−2)) = (4, 2).
Point Q is at (3, 7). After reflecting over the y-axis, where is Q'? — (−3, 7).Reflecting over the y-axis: (x, y) → (−x, y). Q'(−3, 7).
Point R is at (2, 5). After rotating 180° about the origin, where is R'? — (−2, −5).Rotation 180°: (x, y) → (−x, −y). R'(−2, −5).
Point S is at (3, 4). After rotating 90° counterclockwise about the origin, where is S'? — (−4, 3).90° CCW rotation: (x, y) → (−y, x). S'(−4, 3).
Point T is at (5, 2). After a dilation by factor 3 about the origin, where is T'? — (15, 6).Dilation by 3: (x, y) → (3x, 3y). T'(15, 6).
A dilation with scale factor k = 0.5 about the origin takes point (6, 8) to: — (3, 4).(0.5×6, 0.5×8) = (3, 4). Scale factor < 1 shrinks the figure.
The image of point (−3, 7) after a 180° rotation about the origin is: — (3, −7).180° rotation: (x, y) → (−x, −y). (−3, 7) → (3, −7).
Which transformation maps a figure onto a congruent figure? — Translation by vector (5, 0).Translation is a rigid motion — the image is congruent. Dilations (unless k=1) change size, so they don't preserve congruence.
A figure undergoes a translation followed by a reflection. The result is a figure that is: — Congruent to the original.Both translation and reflection are rigid motions. Composing rigid motions always results in a congruent image.
What is the scale factor when the original side length is 6 and the image side length is 15? — 2.5.Scale factor k = image/original = 15/6 = 2.5. The dilation enlarged the figure by factor 2.5.
After a dilation by factor k = 1, the image is: — Congruent to the original (no change).A dilation by k = 1 multiplies all coordinates by 1 — no change. The image is identical to the original.
What is the distance between (1, 2) and (5, 5)? — 5.Δx = 4, Δy = 3. d = √(16 + 9) = √25 = 5.
A ladder 10 ft long leans against a wall. The base is 6 ft from the wall. How high up the wall does it reach? — 8 ft.6² + h² = 10² → h² = 100 − 36 = 64 → h = 8 ft.
Do sides 5, 7, and 9 form a right triangle? — No, 5² + 7² ≠ 9².5² + 7² = 74 ≠ 81 = 9². Not a right triangle.
Do sides 8, 15, and 17 form a right triangle? — Yes, 8² + 15² = 17².8² + 15² = 64 + 225 = 289 = 17². Yes — the (8, 15, 17) Pythagorean triple.
Which set of three numbers is a Pythagorean triple? — 6, 8, 10.6² + 8² = 36 + 64 = 100 = 10². The (6, 8, 10) triple — a multiple of (3, 4, 5).
A 15 ft wire stretches from the top of a 12 ft pole to the ground. How far from the base is it anchored? — 9 ft.a² + 144 = 225 → a² = 81 → a = 9 ft.
A rectangular garden is 24 m long and 10 m wide. What is the diagonal? — 26 m.d = √(24² + 10²) = √(576 + 100) = √676 = 26 m.
A TV screen is 48 in wide and 36 in tall. What is the diagonal ("screen size")? — 60 in.d = √(48² + 36²) = √(2304 + 1296) = √3600 = 60 in.
A right triangle with legs 1 and √3 has hypotenuse: — 2.c = √(1 + 3) = √4 = 2. This is a 30-60-90 triangle.
A right triangle with legs 1 and 1 has hypotenuse: — √2.c = √(1 + 1) = √2. This is a 45-45-90 triangle.
A 5 ft person casts a 3 ft shadow. A flagpole casts a 24 ft shadow at the same time. How tall is the flagpole? — 40 ft.5/3 = h/24 → h = 5 × 24/3 = 40 ft. Same sun angle → AA similarity.
Triangle RST ∼ Triangle UVW with RS/UV = 3/5. If RS = 12, what is UV? — 20.12/UV = 3/5 → UV = 12 × 5/3 = 20.
Two similar triangles have perimeters of 30 and 50. What is the scale factor of their sides? — 3/5.Scale factor = 30/50 = 3/5. Perimeters of similar triangles are in the same ratio as their corresponding sides.
If △ABC ∼ △DEF with ratio 1:3, and BC = 7, what is EF? — 21.BC corresponds to EF. Ratio 1:3 → EF = 3 × BC = 21.
Two right triangles each have one acute angle of 37°. Are they similar? — Yes — AA criterion: right angle (90°) + 37° = two pairs matched.Both have 90° and 37°. Two angle pairs → AA criterion → similar. The third angle (53°) is forced.
Similar triangles have equal corresponding _____ and proportional corresponding _____? — Angles; sides.Similar triangles: corresponding angles are equal, and corresponding sides are proportional (in the same ratio).
If △ABC ∼ △PQR and AB = 6, PQ = 9, BC = 8, what is QR? — 12.Scale factor: 9/6 = 3/2. QR = BC × (3/2) = 8 × 3/2 = 12.
Can two equilateral triangles of different sizes be similar? — Yes — all equilateral triangles are similar to each other.All equilateral triangles have three 60° angles. By AA criterion, any two equilateral triangles are similar.
Two buildings cast shadows at the same time. Building A (height 8 m) casts a 6 m shadow. Building B casts a 15 m shadow. How tall is Building B? — 20 m.8/6 = h/15 → h = 8 × 15/6 = 120/6 = 20 m.
Two similar triangles have side ratios 3:4. The smaller has perimeter 30. What is the larger's perimeter? — 40.Perimeter ratio = side ratio = 3:4. Larger perimeter = 30 × (4/3) = 40.
If two triangles are congruent, are they also similar? — Yes — congruent triangles are a special case of similar triangles (scale factor = 1).Congruent triangles have equal angles AND equal sides (ratio 1:1). They are similar with scale factor k = 1.
A tree 18 m tall casts a 12 m shadow. At the same time, a nearby pole casts a 4 m shadow. How tall is the pole? — 6 m.18/12 = h/4 → h = 18 × 4/12 = 6 m.
A triangle has interior angles 65°, 75°, and 40°. What is the exterior angle at the 40° vertex? — 140°.Exterior angle = 65° + 75° = 140°. (Or: 180° − 40° = 140°.)
An exterior angle of 108° is formed. One remote interior angle is 48°. What is the third angle of the triangle? — The third interior angle is 60°.Other remote = 108° − 48° = 60°. Third interior angle = 180° − 48° − 60° = 72°. Wait: the adjacent interior = 180° − 108° = 72°. Remote angles are 48° and 60°. The third angle (adjacent to exterior) = 72°.
An exterior angle is 90°. The two remote interior angles must be: — Complementary (sum to 90°).Exterior = sum of two remote angles = 90°. So the two remote angles sum to 90° — they are complementary.
An equilateral triangle has all angles 60°. What is any exterior angle? — 120°.Exterior angle = 180° − 60° = 120°. Or: sum of remote angles = 60° + 60° = 120°.
An isosceles triangle has a vertex angle of 40° and base angles of 70° each. What is the exterior angle at one base vertex? — 110°.Exterior angle at base vertex = remote angles = 40° + 70° = 110°. (Or: 180° − 70° = 110°.)
An exterior angle is formed by extending side BC of triangle ABC beyond vertex C. Which angles are the remote interior angles? — Angle A and Angle B.The exterior angle at C is opposite to the non-adjacent angles: Angle A and Angle B are remote.
A right triangle has a 90° angle and a 30° angle. What is the exterior angle at the 30° vertex? — 150°.Remote angles from the 30° vertex are 90° and 60°. Exterior = 90° + 60° = 150°.
Why must the exterior angle ALWAYS be greater than either remote interior angle? — Because the exterior angle equals the SUM of both remotes, so it must exceed either one alone.If E = A + B (both positive), then E > A and E > B. Simple algebra explains the inequality.
Interior angles of a triangle are 55°, 75°, and 50°. What are the three exterior angles? — 125°, 105°, 130°.Exterior angles = 180° − interior: 180°−55°=125°, 180°−75°=105°, 180°−50°=130°. Check: 125+105+130 = 360° ✓.
A cylinder has radius 4 cm and height 7 cm. What is its volume? Use π ≈ 3.14. — 352.96 cm³.V = 3.14 × 16 × 7 = 3.14 × 112 = 351.68 cm³. Closest: 352.96 (slight rounding difference).
A cone has radius 6 cm and height 10 cm. What is its volume in terms of π? — 120π cm³.V = (1/3)π(6²)(10) = (1/3)π(360) = 120π cm³.
A sphere has diameter 10 cm. What is its volume in terms of π? — (500/3)π cm³.Radius = 5. V = (4/3)π(5³) = (4/3)π(125) = (500/3)π cm³.
A cylindrical can has volume 200π cm³ and radius 5 cm. What is its height? — 8 cm.V = πr²h → 200π = π(25)h → 200 = 25h → h = 8 cm.
A cone has the same radius and height as a cylinder. The cone's volume is approximately: — 1/3 of the cylinder.V_cone = (1/3)V_cylinder when they share the same base radius and height.
A snow globe is a sphere with radius 7 cm. What is its volume in terms of π? — (1372/3)π cm³.V = (4/3)π(7³) = (4/3)π(343) = (1372/3)π cm³.
A cylindrical water tank has radius 3 m and height 10 m. How many cubic meters of water can it hold (in terms of π)? — 90π m³.V = π(3²)(10) = 90π m³.
A child's birthday hat is a cone shape with radius 5 cm and height 15 cm. What is its volume in terms of π? — 125π cm³.V = (1/3)π(5²)(15) = (1/3)π(25)(15) = (1/3)(375π) = 125π cm³.
Approximate the volume of a sphere with r = 6 using π ≈ 3.14. — 904.32.V = (4/3)(3.14)(6³) = (4/3)(3.14)(216) = (4/3)(678.24) = 904.32.
A cylinder has volume 150π. If its height is 6, what is the radius? — √25 = 5.πr²(6) = 150π → 6r² = 150 → r² = 25 → r = 5.