Equivalent Forms

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Answer Key & Explanations

Equivalent Forms · Grade 9 · Worksheet 2

1. A right triangle is drawn on a coordinate plane with vertices at (0,0), (8,0), and (0,6). A circle is circumscribed around this 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 hypotenuse of the right triangle. The vertices are (0,0), (8,0), and (0,6). The points (8,0) and (0,6) form the endpoints of the hypotenuse.
   Full step-by-step solution

   Step 1: Identify the hypotenuse of the right triangle. The vertices are (0,0), (8,0), and (0,6). The points (8,0) and (0,6) form the endpoints of the hypotenuse. Step 2: Calculate the length of the hypotenuse using the distance formula: sqrt((8-0)^2 + (0-6)^2) = sqrt(64 + 36) = sqrt(100) = 10 units. Step 3: For a right triangle, the hypotenuse is the diameter of the circumscribed circle. So the diameter is 10 units. Step 4: Calculate the radius: radius = diameter/2 = 10/2 = 5 units. Step 5: Calculate the area of the circle: area = π × radius^2 = 3.14 × 5^2 = 3.14 × 25 = 78.5 square units. The answer is 78.5.

2. (3x² - 5x + 2) + (2x² + 4x - 7) = ? Answer: 5x² - x - 5 Solution: (3x² - 5x + 2) + (2x² + 4x - 7) Since we are adding, the parentheses don't change the signs: 3x² - 5x + 2 + 2x² + 4x - 7 - x² terms: 3x² + 2x² - x terms: -5x + 4x - constant terms: 2 - 7 3x² + 2x² = (3 + 2)x² = 5x² -5x + 4x = (-5 + 4)x = -1x, which is -x 2 - 7 = -5 5x² - x - 5 Final answer: 5x²…
   Full step-by-step solution

   Let's add the two polynomials step by step. We have: (3x² - 5x + 2) + (2x² + 4x - 7) --- **Step 1: Remove parentheses** Since we are adding, the parentheses don't change the signs: 3x² - 5x + 2 + 2x² + 4x - 7 --- **Step 2: Group like terms** - x² terms: 3x² + 2x² - x terms: -5x + 4x - constant terms: 2 - 7 --- **Step 3: Combine x² terms** 3x² + 2x² = (3 + 2)x² = 5x² --- **Step 4: Combine x terms** -5x + 4x = (-5 + 4)x = -1x, which is -x --- **Step 5: Combine constants** 2 - 7 = -5 --- **Step 6: Write the final polynomial** 5x² - x - 5 --- **Final answer:** 5x² - x - 5

3. A right triangle is drawn on a coordinate plane with vertices at (0,0), (4,0), and (4,3). A circle is inscribed within 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) This is a right triangle with the right angle at B = (4,0) because AB is horizontal and BC is vertical.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the triangle** Vertices: A = (0,0) B = (4,0) C = (4,3) This is a right triangle with the right angle at B = (4,0) because AB is horizontal and BC is vertical. So: Side AB = from (0,0) to (4,0) → length = 4 Side BC = from (4,0) to (4,3) → length = 3 Side AC = from (0,0) to (4,3) → length = sqrt((4-0)^2 + (3-0)^2) = sqrt(16 + 9) = sqrt(25) = 5 So triangle sides: 3, 4, 5. --- **Step 2: Recall formula for inradius of a right triangle** For a right triangle with legs a, b and hypotenuse c, the inradius r is: r = (a + b - c) / 2 --- **Step 3: Apply formula** Here a = 3, b = 4, c = 5. r = (3 + 4 - 5) / 2 r = (7 - 5) / 2 r = 2 / 2 r = 1 --- **Step 4: Conclusion** The radius of the inscribed circle is 1. --- **Final answer:** 1

4. Which form of the quadratic y = x² - 14x + 49 reveals the vertex? Explain. Answer: y = (x - 7)² Solution: Start with the standard form: y = x² - 14x + 49. Recognize that 49 = 7² and -14x = -2(7)x, so the expression is a perfect square trinomial. Factor it as (x - 7)².
   Full step-by-step solution

   Step 1: Start with the standard form: y = x² - 14x + 49. Step 2: Recognize that 49 = 7² and -14x = -2(7)x, so the expression is a perfect square trinomial. Step 3: Factor it as (x - 7)². Step 4: The vertex form y = (x - 7)² reveals the vertex at (7, 0), because vertex form is y = (x - h)² + k with vertex (h, k). The answer is y = (x - 7)².

5. Rewrite the expression 9x² + 30x + 25 in a form that reveals it as a perfect square trinomial, then evaluate it when x = 3. Answer: 196 Solution: Recognize the expression 9x² + 30x + 25. The first term 9x² = (3x)², and the last term 25 = 5². Step 2: Check the middle term: 2 × (3x) × 5 = 30x, which matches.
   Full step-by-step solution

   Step 1: Recognize the expression 9x² + 30x + 25. The first term 9x² = (3x)², and the last term 25 = 5². Step 2: Check the middle term: 2 × (3x) × 5 = 30x, which matches. So the expression is a perfect square trinomial: (3x + 5)². Step 3: Substitute x = 3: (3(3) + 5)² = (9 + 5)² = 14² = 196. The answer is 196.

6. A drone is flying along a path modeled by the quadratic function h(t) = -2t² + 12t + 5, where h represents the drone's height in meters above ground level and t represents time in seconds. The drone operator wants to know the maximum height the drone reaches during its flight. What is the maximum height in meters? Answer: 23 Solution: Identify the quadratic function: h(t) = -2t² + 12t + 5 Since the coefficient of t² is negative (-2), the parabola opens downward, meaning the vertex represents the maximum height Find the time at which maximum height occurs using t = -b/(2a), where a = -2 and b = 12 t = -12/(2×(-2)) = -12/(-4) =…
   Full step-by-step solution

   Step 1: Identify the quadratic function: h(t) = -2t² + 12t + 5 Step 2: Since the coefficient of t² is negative (-2), the parabola opens downward, meaning the vertex represents the maximum height Step 3: Find the time at which maximum height occurs using t = -b/(2a), where a = -2 and b = 12 Step 4: t = -12/(2×(-2)) = -12/(-4) = 3 seconds Step 5: Substitute t = 3 into the original function to find the maximum height Step 6: h(3) = -2(3)² + 12(3) + 5 = -2(9) + 36 + 5 = -18 + 36 + 5 = 23 Step 7: The maximum height is 23 meters

7. √(x² - 4x + 4) = ? when x = 5 Answer: 3 Solution: √(x² - 4x + 4) when x = 5. Substitute x = 5 into the expression inside the square root x² - 4x + 4 = (5)² - 4(5) + 4 = 25 - 20 + 4 25 - 20 = 5 5 + 4 = 9 So x² - 4x + 4 = 9 when x = 5.
   Full step-by-step solution

   Let's solve step-by-step. We are given: √(x² - 4x + 4) when x = 5. --- **Step 1: Substitute x = 5 into the expression inside the square root** x² - 4x + 4 = (5)² - 4(5) + 4 = 25 - 20 + 4 --- **Step 2: Simplify the expression inside** 25 - 20 = 5 5 + 4 = 9 So x² - 4x + 4 = 9 when x = 5. --- **Step 3: Take the square root** √(x² - 4x + 4) = √9 --- **Step 4: Interpret the square root** The square root symbol √ means the principal (non-negative) square root. √9 = 3 (since 3 × 3 = 9). --- **Step 5: Check if simplification is possible before substituting** Notice: x² - 4x + 4 = (x - 2)² So √(x² - 4x + 4) = √((x - 2)²) = |x - 2| (since √(a²) = |a|). Now substitute x = 5: |x - 2| = |5 - 2| = |3| = 3. --- **Final Answer:** 3

8. A local park is designing a new rectangular garden with a decorative stone border. The garden's length is 5 meters more than twice its width. If the total area of the garden is 63 square meters, what is the width of the garden in meters? Answer: 4.5 Solution: Let the width of the garden be \( w \) meters. The length is 5 meters more than twice the width, so: length \( l = 2w + 5 \).
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Define the variables** Let the width of the garden be \( w \) meters. The length is 5 meters more than twice the width, so: length \( l = 2w + 5 \). --- **Step 2: Write the area equation** Area of rectangle = length × width Given area = 63 m², so: \[ (2w + 5) \times w = 63 \] --- **Step 3: Expand and rearrange** \[ 2w^2 + 5w = 63 \] Subtract 63 from both sides: \[ 2w^2 + 5w - 63 = 0 \] --- **Step 4: Solve the quadratic equation** We can use the quadratic formula: \[ w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Here \( a = 2 \), \( b = 5 \), \( c = -63 \). First, discriminant \( D = b^2 - 4ac \): \[ D = 5^2 - 4(2)(-63) = 25 + 504 = 529 \] \[ \sqrt{D} = \sqrt{529} = 23 \] --- **Step 5: Apply the formula** \[ w = \frac{-5 \pm 23}{2 \times 2} = \frac{-5 \pm 23}{4} \] Two possible solutions: \[ w = \frac{-5 + 23}{4} = \frac{18}{4} = 4.5 \] \[ w = \frac{-5 - 23}{4} = \frac{-28}{4} = -7 \] --- **Step 6: Choose the valid solution** Width cannot be negative, so \( w = 4.5 \) meters. --- **Final answer:** 4.5

9. 2x² - 8x + 6 = ? when x = 3 Answer: 0 Solution: We are given the expression: 2x² - 8x + 6, and x = 3. Substitute x = 3 into the expression. This means we replace every x with 3: 2*(3)² - 8*(3) + 6 Calculate the exponent first (order of operations: PEMDAS).
   Full step-by-step solution

   We are given the expression: 2x² - 8x + 6, and x = 3. Step 1: Substitute x = 3 into the expression. This means we replace every x with 3: 2*(3)² - 8*(3) + 6 Step 2: Calculate the exponent first (order of operations: PEMDAS). (3)² = 9 So the expression becomes: 2*9 - 8*3 + 6 Step 3: Perform the multiplications. 2*9 = 18 8*3 = 24 So now we have: 18 - 24 + 6 Step 4: Perform the addition and subtraction from left to right. 18 - 24 = -6 Then -6 + 6 = 0 Step 5: Conclusion. The value of 2x² - 8x + 6 when x = 3 is 0.