Equivalent Forms

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Answer Key & Explanations

Equivalent Forms · Grade 9 · Worksheet 1

1. A right triangle is drawn on a coordinate plane with vertices at (0,0), (6,0), and (6,8). A circle is circumscribed around the triangle such that all three vertices lie on the circle's circumference. What is the area of the circumscribed circle? (Use π = 3.14) Answer: 78.5 Solution: Identify the triangle's sides. The vertices are (0,0), (6,0), and (6,8). From (0,0) to (6,0): length = 6 (horizontal side).
   Full step-by-step solution

   Step 1: Identify the triangle's sides. The vertices are (0,0), (6,0), and (6,8). From (0,0) to (6,0): length = 6 (horizontal side). From (6,0) to (6,8): length = 8 (vertical side). From (0,0) to (6,8): length = sqrt((6-0)^2 + (8-0)^2) = sqrt(36 + 64) = sqrt(100) = 10. So the triangle has sides 6, 8, 10 — a right triangle with hypotenuse 10. Step 2: Circumscribed circle property for a right triangle. For a right triangle, the hypotenuse is the diameter of the circumscribed circle. So the diameter of the circle = length of hypotenuse = 10. Step 3: Find the radius. Radius r = diameter / 2 = 10 / 2 = 5. Step 4: Area of the circle. Area = π × r^2 = 3.14 × (5^2) = 3.14 × 25. Step 5: Multiply. 3.14 × 25 = 78.5. Final answer: The area of the circumscribed circle is 78.5.

2. (2x² - 5x + 3)(x - 2) = ? Answer: 2x³ - 9x² + 13x - 6 Solution: Multiply each term in (2x² - 5x + 3) by each term in (x - 2) 2x² × x = 2x³ 2x² × (-2) = -4x² (-5x) × x = -5x² (-5x) × (-2) = 10x 3 × x = 3x 3 × (-2) = -6 Combine all terms: 2x³ + (-4x² - 5x²) + (10x + 3x) - 6 Simplify: 2x³ - 9x² + 13x - 6 The answer is 2x³ - 9x² + 13x - 6.
   Full step-by-step solution

   Step 1: Multiply each term in (2x² - 5x + 3) by each term in (x - 2) Step 2: 2x² × x = 2x³ Step 3: 2x² × (-2) = -4x² Step 4: (-5x) × x = -5x² Step 5: (-5x) × (-2) = 10x Step 6: 3 × x = 3x Step 7: 3 × (-2) = -6 Step 8: Combine all terms: 2x³ + (-4x² - 5x²) + (10x + 3x) - 6 Step 9: Simplify: 2x³ - 9x² + 13x - 6 The answer is 2x³ - 9x² + 13x - 6.

3. Charlotte is analyzing the trajectory of a drone during a search-and-rescue operation. The drone's height in meters above the ground at time t seconds is given by the quadratic function h(t) = -4t² + 32t + 2. Charlotte needs to rewrite this function in vertex form to determine the maximum height the drone reaches and when it occurs. Which equivalent form of the function reveals these properties, and what are the coordinates of the vertex? Answer: h(t) = -4(t - 4)² + 66; vertex (4, 66); maximum height 66 meters at t = 4 seconds Solution: Start with h(t) = -4t² + 32t + 2. Factor out -4 from the first two terms: h(t) = -4(t² - 8t) + 2. Complete the square inside the parentheses: Take half of -8, which is -4, and square it to get 16.
   Full step-by-step solution

   Step 1: Start with h(t) = -4t² + 32t + 2. Step 2: Factor out -4 from the first two terms: h(t) = -4(t² - 8t) + 2. Step 3: Complete the square inside the parentheses: Take half of -8, which is -4, and square it to get 16. Add and subtract 16 inside: t² - 8t + 16 - 16 = (t - 4)² - 16. Step 4: Substitute back: h(t) = -4[(t - 4)² - 16] + 2. Step 5: Distribute -4: h(t) = -4(t - 4)² + 64 + 2. Step 6: Simplify: h(t) = -4(t - 4)² + 66. Step 7: Vertex form is h(t) = -4(t - 4)² + 66. The vertex is (4, 66). Since a = -4 is negative, the parabola opens downward, so the vertex is a maximum. Step 8: The drone reaches its maximum height of 66 meters at t = 4 seconds. The answer is h(t) = -4(t - 4)² + 66, vertex (4, 66).

4. 2^(x+1) - 2^(x-1) = 24 Answer: 4 Solution: Factor out 2^(x-1) from both terms: 2^(x-1)(2^2 - 1) = 24 Simplify inside parentheses: 2^(x-1)(4 - 1) = 24 Calculate: 2^(x-1)(3) = 24 Divide both sides by 3: 2^(x-1) = 8 Rewrite 8 as a power of 2: 2^(x-1) = 2^3 Set exponents equal: x - 1 = 3 Solve for x: x = 4 The answer is 4.
   Full step-by-step solution

   Step 1: Factor out 2^(x-1) from both terms: 2^(x-1)(2^2 - 1) = 24 Step 2: Simplify inside parentheses: 2^(x-1)(4 - 1) = 24 Step 3: Calculate: 2^(x-1)(3) = 24 Step 4: Divide both sides by 3: 2^(x-1) = 8 Step 5: Rewrite 8 as a power of 2: 2^(x-1) = 2^3 Step 6: Set exponents equal: x - 1 = 3 Step 7: Solve for x: x = 4 The answer is 4.

5. Hana is analyzing the trajectory of a drone for her engineering project. The height of the drone above the ground, in meters, is given by the function h(t) = -2t² + 20t - 42, where t is the time in seconds. Hana needs to find the times when the drone is at ground level (height = 0). She has two equivalent forms of the function: h(t) = -2(t - 3)(t - 7) and h(t) = -2(t - 5)² + 8. Which form should she use to directly determine the times when the drone hits the ground, and what are those times? Answer: t = 3 seconds and t = 7 seconds Solution: The drone is at ground level when h(t) = 0. We need to find the values of t that satisfy -2t² + 20t - 42 = 0.
   Full step-by-step solution

   Step 1: The drone is at ground level when h(t) = 0. We need to find the values of t that satisfy -2t² + 20t - 42 = 0. Step 2: Hana has two equivalent forms: the factored form h(t) = -2(t - 3)(t - 7) and the vertex form h(t) = -2(t - 5)² + 8. Step 3: The factored form directly shows the zeros (roots) of the function. When a product equals zero, at least one factor must be zero. So set (t - 3)(t - 7) = 0. Step 4: Solve: t - 3 = 0 gives t = 3, and t - 7 = 0 gives t = 7. Step 5: Therefore, the drone hits the ground at t = 3 seconds and t = 7 seconds. The answer is t = 3 seconds and t = 7 seconds.

6. A right triangle is drawn on a coordinate plane with vertices at (0,0), (4,0), and (4,3). A circle is inscribed inside this triangle, tangent to all three sides. What is the radius of this inscribed circle? Answer: 1 Solution: A = (0,0), B = (4,0), C = (4,3). Side AB is along the x-axis from (0,0) to (4,0), length = 4. Side BC is vertical from (4,0) to (4,3), length = 3.
   Full step-by-step solution

   Step 1: Understand the triangle and circle We have a right triangle with vertices: A = (0,0), B = (4,0), C = (4,3). Side AB is along the x-axis from (0,0) to (4,0), length = 4. Side BC is vertical from (4,0) to (4,3), length = 3. Side AC is the hypotenuse from (0,0) to (4,3). Step 2: Find the hypotenuse length Using the distance formula between (0,0) and (4,3): AC = sqrt((4-0)^2 + (3-0)^2) = sqrt(16 + 9) = sqrt(25) = 5. So sides: 3, 4, 5. Step 3: Recall formula for inradius of a right triangle For a right triangle with legs a and b and hypotenuse c, the inradius r is: r = (a + b - c) / 2. Step 4: Apply the formula Here a = 3, b = 4, c = 5. r = (3 + 4 - 5) / 2 = (7 - 5) / 2 = 2 / 2 = 1. Step 5: Interpret the result The inscribed circle is tangent to all three sides of the triangle at a perpendicular distance r = 1 from each side. So the radius is 1. Final answer: 1

7. Mason graphs the quadratic function f(x) = x^2 - 16x + 60 on a coordinate plane. The parabola opens upward and crosses the x-axis at two points. Which equivalent form of the function reveals the x-coordinate of the vertex, and what is that x-coordinate? Answer: 8 Solution: Start with f(x) = x^2 - 16x + 60. To complete the square, take half of the coefficient of x, which is -16/2 = -8, and square it: (-8)^2 = 64. Add and subtract 64 inside the expression: f(x) = (x^2 - 16x + 64) + 60 - 64.
   Full step-by-step solution

   Step 1: Start with f(x) = x^2 - 16x + 60. Step 2: To complete the square, take half of the coefficient of x, which is -16/2 = -8, and square it: (-8)^2 = 64. Step 3: Add and subtract 64 inside the expression: f(x) = (x^2 - 16x + 64) + 60 - 64. Step 4: Simplify the perfect square trinomial: (x^2 - 16x + 64) = (x - 8)^2. Step 5: Combine the constants: 60 - 64 = -4. Step 6: Vertex form is f(x) = (x - 8)^2 - 4. Step 7: The vertex is at (8, -4), so the x-coordinate of the vertex is 8. The answer is 8.