Statistics and Probability
A class survey: 12 choose circle, 8 choose square, 5 choose triangle. What is the probability a randomly chosen student prefers a circle?
12/(12+8+5) = 12/25 = 0.48
Define a circle in a plane.
The set of all points in a plane at a fixed distance (radius) from a fixed point (center).
When a transversal crosses two parallel lines, what relationship do alternate interior angles have?
Alternate interior angles are congruent.
Effect of dilation with scale factor 1/2 on side lengths.
All side lengths are multiplied by 1/2 (halved).
Name one congruence criterion.
Example: Side-Angle-Side (SAS)
A spinner has 4 equal sectors. What is probability of landing on sector A? Give fraction and percent.
1/4 = 25%
Difference between radius and diameter.
Diameter = 2 × radius.
Slope of a line perpendicular to a line with slope 2?
-1/2
Rotation 90° counterclockwise about origin sends (3, −2) to what?
(−y, x) so (3, −2) → (2, 3).
Why are base angles of an isosceles triangle congruent?
In isosceles triangle with equal sides AB = AC, triangles formed by dropping altitude from A to BC are congruent by SSS or by reflecting, giving base angles at B and C equal.
Quiz scores: 70, 80, 85, 90, 95. Find mean and median.
Mean = (70+80+85+90+95)/5 = 420/5 = 84. Median = 85
Define tangent to a circle.
A line in the plane that intersects the circle at exactly one point and is perpendicular to the radius at the point of tangency.
Equation of line through (1,2) with slope 3.
y − 2 = 3(x − 1) → y = 3x − 1
Reflection across the line y = x: image of (a,b)?
(b,a)
Two right triangles with hypotenuse 10 and one leg 6 are congruent? Use HL.
Hypotenuse = 10 and one leg = 6 match in both triangles, so by Hypotenuse-Leg (HL) congruence for right triangles, triangles are congruent.
Game: win $200 for correct, cost $50 to play. Probability correct = 0.6. What is expected value (net) per play?
EV = 0.6*(200 − 50) + 0.4*(−50) = 0.6*150 − 20 = 90 − 20 = $70 (Alternate interpretation: if cost is paid regardless and prize is $200 only when correct, net outcomes are +150 with prob 0.6 and −50 with prob 0.4; EV = $70.)
Why are all circles similar?
Similarity means one shape is a scaled copy of another. Any two circles with radii r1 and r2 can be mapped by a dilation centered at their centers with scale factor r2/r1 (plus a translation), so corresponding angles and proportional distances hold. Therefore all circles are similar.
Explain why any point on a perpendicular bisector of segment AB is equidistant from A and B (short reasoning).
Let P be on perpendicular bisector of AB. Then PB and PA are radii of circles centered at P passing through A and B; triangles P A M and P B M (M midpoint) are congruent by SAS, so PA = PB.
Give a sequence of transformations mapping triangle ABC to congruent triangle A'B'C' (example). Problem: Suppose A(1,0), B(3,0), C(1,2). Map to A'(−1,0), B'(−3,0), C'(−1,2). Sequence: reflect across y-axis (x→−x). This single reflection maps ABC to A'B'C' (congruent).
Reflection across the y axis
Given A(0,0), B(4,0), C(0,3). Find side lengths and determine if congruent to A'(1,1), B'(5,1), C'(1,4).
For original triangle, AB = 4, AC = 3, BC = 5 (3-4-5). For prime triangle, A'B' distance = 4, A'C' = 3, B'C' = 5. Same side lengths → triangles are congruent (translation of (1,1) shift).
Flip a fair coin 4 times. Probability of exactly two heads?
C(4,2)/2^4 = 6/16 = 3/8
Equation of circle centered at (2, −3) with radius 5.
(x−2)2+(y+3)2=25
In triangle ABC, a line parallel to AB intersects AC at D and BC at E. If AC = 9, AD = 4, and DC = 5, find AE:EC ratio and scale factor between triangle ADE and ABC; also find AE if BC = 12.
Since AD/AC = 4/9, scale factor smaller/larger = 4/9. For AE on BC, corresponding division: AE/BC = AD/AC = 4/9 → AE = (4/9)*12 = 48/9 = 16/3 ≈ 5.333. AE:EC = (16/3) : (12 − 16/3) = (16/3) : (20/3) = 16:20 = 4:5.
Symmetries of a regular pentagon.
Rotational symmetry of order 5 (rotations by multiples of 72°) and 5 lines of reflectional symmetry (through each vertex and midpoint of opposite side).
Describe demonstrating congruence by a translation and a reflection (constructive).
To map triangle ABC to A'B'C', first translate so one vertex aligns; then reflect across appropriate line to align orientation; if after these rigid motions all three vertices coincide, triangles congruent. (Example: translate by vector from A to A', then reflect across perpendicular bisector of image of AB and A'B' to align B.)