Coordinate Proofs

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Answer Key & Explanations

Coordinate Proofs · Grade 10 · Worksheet 2

1. Prove that triangle ABC with vertices A(1,1), B(5,1), C(3,7) is isosceles using coordinate geometry. Answer: Triangle ABC is isosceles because AB = BC = 4 units, while AC = 2√10 units Solution: An isosceles triangle has at least two sides of equal length. Using the distance formula, we can calculate the lengths of all three sides and compare them. The distance formula is d = √[(x₂-x₁)² + (y₂-y₁)²].
2. Prove that triangle Hana with vertices H(0,0), A(6,0), N(2,4) is isosceles using coordinate geometry. Answer: The triangle is isosceles because sides HN and AN both have length sqrt(20) Solution: To prove a triangle is isosceles using coordinate geometry, calculate the distances between all pairs of vertices using the distance formula. If two sides have equal lengths, the triangle is isosceles. The distance formula is d = sqrt((x₂-x₁)² + (y₂-y₁)²).
3. Ava is designing a triangular garden plot with vertices at A(1,1), B(6,1), and C(4,6). She wants to prove it's an isosceles triangle using coordinate geometry. Which statement correctly proves the triangle is isosceles? Answer: A. AB = AC Solution: In coordinate geometry, we can prove a triangle is isosceles by showing that two of its sides have equal lengths using the distance formula. The distance between two points (x1,y1) and (x2,y2) is calculated as sqrt((x2-x1)^2 + (y2-y1)^2).
   Full step-by-step solution

   In coordinate geometry, we can prove a triangle is isosceles by showing that two of its sides have equal lengths using the distance formula. The distance between two points (x1,y1) and (x2,y2) is calculated as sqrt((x2-x1)^2 + (y2-y1)^2). For example, if you had points D(0,0), E(3,0), and F(1,4), you would calculate DE = sqrt((3-0)^2 + (0-0)^2) = 3, DF = sqrt((1-0)^2 + (4-0)^2) = sqrt(17), and EF = sqrt((1-3)^2 + (4-0)^2) = sqrt(20). Comparing these lengths would help you determine if the triangle is isosceles.

4. Prove that quadrilateral Isabella with vertices I(8,2), S(12,6), A(16,2), and B(12,10) is a rhombus using coordinate geometry. Answer: All four sides have length sqrt(32) and the diagonals are perpendicular with slopes 0 and undefined Solution: A rhombus is a special quadrilateral where all four sides are congruent and the diagonals are perpendicular bisectors of each other.
   Full step-by-step solution

   A rhombus is a special quadrilateral where all four sides are congruent and the diagonals are perpendicular bisectors of each other. The distance formula helps verify equal side lengths, while the slope formula can show perpendicular diagonals. Remember that perpendicular lines have slopes that are negative reciprocals of each other.

5. Aroha is creating a triangular banner for a school festival. She places the vertices at coordinates A(9, 4), B(15, 12), and C(21, 4). She wants to prove the banner is isosceles. What is the length of the two equal sides? Round your answer to the nearest tenth if needed. Answer: 10 Solution: Calculate distance AB using distance formula: sqrt((15-9)^2 + (12-4)^2) = sqrt(6^2 + 8^2) = sqrt(36 + 64) = sqrt(100) = 10 Calculate distance BC: sqrt((21-15)^2 + (4-12)^2) = sqrt(6^2 + (-8)^2) = sqrt(36 + 64) = sqrt(100) = 10 Calculate distance AC: sqrt((21-9)^2 + (4-4)^2) = sqrt(12^2 + 0^2) =…
   Full step-by-step solution

   Step 1: Calculate distance AB using distance formula: sqrt((15-9)^2 + (12-4)^2) = sqrt(6^2 + 8^2) = sqrt(36 + 64) = sqrt(100) = 10 Step 2: Calculate distance BC: sqrt((21-15)^2 + (4-12)^2) = sqrt(6^2 + (-8)^2) = sqrt(36 + 64) = sqrt(100) = 10 Step 3: Calculate distance AC: sqrt((21-9)^2 + (4-4)^2) = sqrt(12^2 + 0^2) = sqrt(144) = 12 Step 4: Compare the side lengths: AB = 10, BC = 10, AC = 12 Step 5: Since AB = BC = 10, the triangle is isosceles with two equal sides of length 10. The answer is 10.

6. Prove that triangle Aroha with vertices A(1, 3), R(5, 7), and H(9, 3) is isosceles using coordinate geometry. Answer: AR = RH = 4√2, so triangle Aroha is isosceles Solution: An isosceles triangle has at least two sides of equal length. By calculating the distances between each pair of vertices using the distance formula, you can determine if this condition is met.
7. Prove that triangle Aroha with vertices A(9,2), B(15,2), C(12,10) is isosceles using coordinate geometry. Answer: Triangle Aroha is isosceles because AB = 6, AC = sqrt(73), BC = sqrt(73), and AC = BC. Solution: An isosceles triangle has at least two sides of equal length. Using the distance formula d = sqrt((x₂-x₁)² + (y₂-y₁)²), you can calculate the lengths of all three sides and compare them to determine if any two are equal.
8. Tane is designing a triangular garden plot with vertices at A(3, 5), B(9, 11), and C(15, 5). He wants to prove that the triangle is isosceles by showing that two sides have equal length. Which pair of sides should Tane compare to prove this? Answer: D. AB and AC Solution: When proving a triangle is isosceles using coordinate geometry, you calculate the lengths of all three sides using the distance formula: distance = sqrt((x2-x1)^2 + (y2-y1)^2).
   Full step-by-step solution

   When proving a triangle is isosceles using coordinate geometry, you calculate the lengths of all three sides using the distance formula: distance = sqrt((x2-x1)^2 + (y2-y1)^2). An isosceles triangle has exactly two sides of equal length. In a different scenario, if you had points at (1,1), (5,1), and (3,7), you would calculate each side length to determine which two are equal.