Pythagorean Theorem
A right triangle has legs of 3 cm and 4 cm. What is the length of the hypotenuse?
Answer: 5 cm (3² + 4² = 25; √25 = 5)
Is 0.375 rational or irrational? Explain.
Answer: Rational — it terminates (= 3/8)
A bag has 4 red, 3 blue, and 3 green marbles. What is the theoretical probability of drawing a blue marble?
Answer: 3/10 or 30%
Write 0.00047 in scientific notation.
Answer: 4.7 × 10⁻⁴
Triangle ABC is translated 3 units right and 4 units down. If A is at (1, 2), where is A'?
Answer: (4, −2)
A right triangle has a hypotenuse of 13 and one leg of 5. What is the length of the other leg?
Answer: 12 (13² − 5² = 144; √144 = 12)
Classify each: √9, √10, 0.333…, π
Answer: √9 = rational (= 3); √10 = irrational; 0.333… = rational (= 1/3); π = irrational
Students can choose between chicken tacos or beef tacos, and queso or guacamole.
How large is the sample space?
Answer: 4. Chicken and Queso, Chicken and Guacamole, Beef and Queso, Beef and Guacamole.
Simplify: x⁵ · x³
Answer: x⁸ (Product of Powers Rule)
Point P is at (3, −5). What are the coordinates of P' after a reflection over the x-axis?
Answer: (3, 5)
Is a triangle with sides 7, 10, and 12 a right triangle? Show how you know.
Answer: No — 7² + 10² = 149 ≠ 144 = 12²
Between which two consecutive integers does √50 lie?
Answer: Between 7 and 8 (7² = 49, 8² = 64)
Based on a survey of 40 students, 15 prefer soccer. In a school of 600 students, how many would you predict prefer soccer?
Answer: 15/40 × 600 = 225 students
Evaluate: (3 × 10⁴) × (2 × 10³)
Answer: 6 × 10⁷
Point Q is at (4, 2). It is dilated by a scale factor of 1/2 with the center of dilation at the origin. What are the coordinates of Q'?
Answer: (2, 1)
Two points are located at (1, 2) and (7, 10) on a coordinate plane. What is the distance between them? Leave your answer in simplest radical form.
Answer: 10 units (√(36 + 64) = √100 = 10)
Place these numbers in order from least to greatest: √5, 2.1, 7/3, √3
Answer: √3 ≈ 1.73, √5 ≈ 2.24, 7/3 ≈ 2.33, 2.1 — order: √3, 2.1, √5, 7/3
Two fair number cubes (1–6) are rolled. What is the probability that you roll a 5 and then a 6.
Answer: 1/36 ≈ 2.7%
Simplify: (x³)⁴
Answer: x¹² (Power of a Product Rule)
A figure is rotated 90° clockwise about the origin. If one vertex is at (−3, 4), where does it land?
Answer: (4, 3) — rule: (x, y) → (y, −x)
A 15-foot ladder leans against a wall. Its base is 6 feet from the wall. How high up the wall does the ladder reach? Round to the nearest tenth.
Answer: ≈ 13.7 feet (√(225 − 36) = √189 ≈ 13.748)
You are playing a card game. First, you draw a card from a deck of 52 cards. Then, you flip a coin. Finally, you roll a cube.
What is the sample space of the game?
Answer: 52*2*4 = 624 possibilities.
The distance from Earth to the Moon is about 3.84 × 10^8 mm. The diameter of a human hair is about 7 × 10⁻⁵ m. How many times greater is the Earth-Moon distance than the diameter of a human hair? Express your answer in scientific notation.
Answer: (3.84×10⁸) ÷ (7×10⁻⁵) ≈ 5.49×10¹²
Rectangle ABCD with A(2, 2), B(6, 2), C(6, 4), D(2, 4) is dilated by scale factor 3 centered at the origin. What is the area of the image A'B'C'D'?
Answer: Area of original = 8; scale factor² = 9; image area = 72 square units