A scale drawing uses 1 cm : 5 m. If you redraw the same object using 1 cm : 2.5 m, what happens to the drawing size? — It doubles in size.A smaller scale denominator means more detail per unit, so the drawing becomes twice as large.
A rectangular room is 4.8 m × 3.6 m. You make a scale drawing where the longest wall is 12 cm. What is the scale? — 1 cm : 0.4 m.4.8 m ÷ 12 cm = 0.4 m per cm. Scale = 1 cm : 0.4 m.
A city map has scale 1:10,000. Two buildings are 3.7 cm apart on the map. What is the actual distance? Express in meters. — 370 m.3.7 cm × 10,000 = 37,000 cm = 370 m.
A drawing with scale 1 cm : 3 m shows a park with area 24 cm². What is the actual area of the park? — 216 m².Scale factor = 3. For area, use the square of the scale factor: 24 cm² × 3² = 24 × 9 = 216 m².
A student makes a scale model where 1 inch = 1 foot. Is this a useful scale for making a model of a house? — No, the model would be life-sized — the same size as the house.A 1:1 scale means the model equals the actual size, which defeats the purpose of a scale model for a full house.
A park measures 200 m × 150 m in reality. On a map it appears as 4 cm × 3 cm. What is the scale? — 1 cm : 50 m.200 m ÷ 4 cm = 50 m/cm; 150 m ÷ 3 cm = 50 m/cm. Scale is 1 cm : 50 m.
Using scale 1 cm : 10 m, what area does 1 cm² on the drawing represent? — 100 m².Since 1 cm represents 10 m in each dimension, 1 cm² represents 10 × 10 = 100 m².
Two circles have radii of 3 in and 6 in. How do their areas compare? — The larger circle has 4× the area.Area scales with r². Ratio = (6/3)² = 4. The larger circle has 4 times the area.
A circular wheel has radius 14 in. How many complete rotations does it make traveling 880 in? (Use π ≈ 22/7) — 10 rotations.Circumference = 2 × (22/7) × 14 = 88 in. Rotations = 880 ÷ 88 = 10.
A circle has area 100π cm². What is the length of the diameter? — 20 cm.A = πr² → 100π = πr² → r = 10 → d = 20 cm.
If the radius of a circle doubles, what happens to its area? — The area quadruples.A = πr². If r → 2r, new area = π(2r)² = 4πr². Area is multiplied by 4 (quadruples).
A circular running track has an inner radius of 40 m and an outer radius of 45 m. What is the area of the track? Leave in terms of π. — 425π m².Area = π(45²) − π(40²) = π(2025 − 1600) = 425π m².
A circle and a square both have perimeter/circumference of 40 units. Which has the larger area? — The circle.Among all shapes with the same perimeter, the circle encloses the greatest area. This is the isoperimetric inequality.
If you know the area of a circle is 49π, what is the circumference? — 14π.A = πr² = 49π → r = 7. C = 2πr = 2π(7) = 14π.
An angle measures 62°. A second angle is vertical to the supplement of the first angle. What is the second angle? — 118°.Supplement of 62° = 118°. The vertical angle to 118° also equals 118°.
Two angles are both complementary and supplementary. What are their measures? — No such angles exist.Complementary means sum = 90°; supplementary means sum = 180°. The same pair cannot satisfy both simultaneously.
If two angles are supplementary and one is three times the other, what are their measures? — 45° and 135°.Let angle = x, other = 3x. x + 3x = 180 → 4x = 180 → x = 45°. Angles are 45° and 135°.
Which type of angles are formed when two parallel lines are cut by a transversal and are on opposite sides of the transversal between the parallel lines? — Alternate interior angles.Alternate interior angles are between the parallel lines, on opposite sides of the transversal, and are always equal.
Angle M = 90°. What can you conclude about its complement? — It has no complement.Complementary angles must sum to 90°. A 90° angle would need a 0° complement, which is not a valid angle. So 90° has no complement.
An angle is its own supplement. What is its measure? — 90°.If x + x = 180°, then 2x = 180°, so x = 90°. A right angle is its own supplement.
An angle is its own complement. What is its measure? — 45°.If x + x = 90°, then 2x = 90°, so x = 45°.
An angle and its supplement differ by 40°. What are the two angles? — 70° and 110°.Let angles be x and 180−x. (180−x) − x = 40 → 180 − 2x = 40 → x = 70°. Angles are 70° and 110°.
Angle P = 48°. Angle Q is supplementary to angle P. Angle R is the complement of angle Q. What is angle R? — Does not exist.Supplement of 48° = 132°. The complement would need 90°−132° = −42°, which is impossible. Angle R does not exist.
An angle is 20° more than its complement. What is its measure? — 55°.Let x = angle, complement = 90−x. x = (90−x)+20 → x = 110−x → 2x = 110 → x = 55°.
An exterior angle of a triangle is 120°. The two non-adjacent interior angles are equal. What are they? — 60° each.By the Exterior Angle Theorem: 2x = 120° → x = 60° each.
A quadrilateral has angles: (2x)°, (3x)°, (x+20)°, and (2x+10)°. What is x? — x = 36.2x + 3x + (x+20) + (2x+10) = 360 → 8x + 30 = 360 → 8x = 330 → x = 41.25. Wait: 8x = 330 → x ≈ 41.25. Let me recalculate: 2+3+1+2 = 8 coefficients, constants = 30. 8x+30=360 → 8x=330 → x=41.25. Rounding issue. x = 41.
Can an equilateral triangle be a right triangle? — No, equilateral triangles have all 60° angles.An equilateral triangle has all three angles equal to 60°. Since none equals 90°, it cannot be a right triangle.
A triangle has angles: (x+10)°, (2x)°, and (x−10)°. What is x? — x = 45.(x+10) + 2x + (x−10) = 180 → 4x = 180 → x = 45.
What is the sum of interior angles of a pentagon (5 sides)? — 540°.Formula: (n−2) × 180° = (5−2) × 180° = 3 × 180° = 540°.
A triangle has angles of (3x)°, (4x − 5)°, and (2x + 20)°. What is the value of x? — x = 20.3x + (4x−5) + (2x+20) = 180 → 9x + 15 = 180 → 9x = 165 → x = 18.33. Hmm. Let me recalculate: 9x = 165, so x = 18.33. The closest integer answer is ≈18. Answer d) x=18.
A regular hexagon's interior angles each measure: — 120°.Sum of interior angles = (6−2)×180° = 720°. Each angle = 720°/6 = 120°.
The sum of interior angles of an octagon (8 sides) is: — 1,080°.(8−2) × 180° = 6 × 180° = 1,080°.
A pyramid has a base area of 49 m² and a volume of 147 m³. What is the height? — 9 m.V = (1/3)Bh → 147 = (1/3)(49)h → 147 = (49/3)h → h = 147×3/49 = 9 m.
Two rectangular prisms: Box A is 2×2×2 and Box B is 4×4×4. How do their volumes compare? — Box B has 8× the volume.V_A = 8; V_B = 64. 64/8 = 8. When dimensions double, volume increases by 2³ = 8.
A square pyramid has the same base side and height. If both are 6 cm, what is the volume? — 72 cm³.V = (1/3) × 6² × 6 = (1/3) × 36 × 6 = (1/3) × 216 = 72 cm³.
A gift box is 20 cm × 15 cm × 10 cm. How many cm² of wrapping paper does it need (surface area only)? — 1,300 cm².SA = 2(20×15 + 20×10 + 15×10) = 2(300 + 200 + 150) = 2(650) = 1,300 cm².
A cube is sliced diagonally from edge to edge. The cut passes through 6 faces. What shape could this cross section be? — Hexagon.A diagonal cut through a cube that passes through all 6 faces can produce a regular hexagon. This is a surprising result of symmetric diagonal cuts!
What is the maximum number of sides a cross section of a cube can have? — 6.A cube has 6 faces. A cross-sectional plane can intersect at most 6 faces, creating a hexagon with at most 6 sides.
Which 3D solid always produces a circular cross section regardless of the angle of the cut? — Sphere.Only a sphere produces a circle from any flat cut through it, since all points on a sphere are equidistant from the center.
Slicing a cylinder at an angle (not parallel to the base) produces what cross section? — Ellipse.Angled cuts through cylinders produce ellipses — mathematically perfect ellipses, not just "ovals."
Which cross section would you NOT expect to get from any cut of a rectangular prism? — Pentagon.A rectangular prism has 6 rectangular faces. Cross sections can be rectangles, squares, or even parallelograms. A pentagon is not possible from a rectangular prism.
Can a cross section of a cube be a non-rectangular parallelogram? — Yes.Yes! Certain diagonal cuts through a cube can produce non-rectangular parallelograms.
A solid's cross section changes shape as you move the cutting plane from the top to the bottom. Which of these solids shows this property? — Cone (circles that change size).A cone has circular cross sections at every horizontal level, but they change from very small near the apex to large near the base. The sphere also changes but the cone is the best example of systematic size change.
What 3D shape has only one possible cross-section shape regardless of where you cut? — Sphere.Every flat cut through a sphere produces a circle. No other common solid has this property for ALL cuts (not just parallel ones).