Coordinate Proofs

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Coordinate Proofs · Grade 10 · Worksheet 1

1. Prove that triangle ABC with vertices A(1, 3), B(5, 7), and C(9, 3) is isosceles using coordinate geometry. Answer: Triangle ABC is isosceles because AB = BC = √32 Solution: An isosceles triangle has at least two sides of equal length. The distance formula can be used to determine if any two sides are equal.
2. Aroha is proving that a triangular garden with vertices at (12, 9), (24, 18), and (36, 9) is isosceles. She needs to find the length of the longest side to determine how much fencing material to order for that side. What is the length of the longest side of this triangular garden? Answer: 24 Solution: Calculate distance between (12, 9) and (24, 18) Distance = sqrt((24-12)^2 + (18-9)^2) = sqrt(12^2 + 9^2) = sqrt(144 + 81) = sqrt(225) = 15 Calculate distance between (24, 18) and (36, 9) Distance = sqrt((36-24)^2 + (9-18)^2) = sqrt(12^2 + (-9)^2) = sqrt(144 + 81) = sqrt(225) = 15 Calculate…
   Full step-by-step solution

   Step 1: Calculate distance between (12, 9) and (24, 18) Distance = sqrt((24-12)^2 + (18-9)^2) = sqrt(12^2 + 9^2) = sqrt(144 + 81) = sqrt(225) = 15 Step 2: Calculate distance between (24, 18) and (36, 9) Distance = sqrt((36-24)^2 + (9-18)^2) = sqrt(12^2 + (-9)^2) = sqrt(144 + 81) = sqrt(225) = 15 Step 3: Calculate distance between (12, 9) and (36, 9) Distance = sqrt((36-12)^2 + (9-9)^2) = sqrt(24^2 + 0^2) = sqrt(576) = 24 Step 4: Compare the three distances: 15, 15, and 24 The longest side is 24.

3. Prove that triangle Emma with vertices E(3,1), M(9,1), M(6,7) is isosceles using coordinate geometry. Answer: The triangle is isosceles because EM = MM = sqrt(45) Solution: An isosceles triangle has at least two sides of equal length. The distance formula helps determine side lengths from coordinates.
4. Prove that quadrilateral Aroha with vertices A(10,4), B(16,4), C(14,10), D(12,10) is an isosceles trapezoid. Answer: To prove quadrilateral Aroha is an isosceles trapezoid, we need to show: 1) One pair of parallel sides, 2) Non-parallel sides are equal in length. First, calculate slopes: AB: (4-4)/(16-10) = 0/6 = 0, CD: (10-10)/(12-14) = 0/-2 = 0. Since AB and CD both have slope 0, they are parallel. Now calculate lengths of non-parallel sides: AD = sqrt((12-10)^2 + (10-4)^2) = sqrt(4 + 36) = sqrt(40), BC = sqrt((14-16)^2 + (10-4)^2) = sqrt(4 + 36) = sqrt(40). Since AD = BC, the non-parallel sides are equal. Therefore, quadrilateral Aroha is an isosceles trapezoid. Solution: An isosceles trapezoid has exactly one pair of parallel sides (bases) and the non-parallel sides (legs) are congruent.
   Full step-by-step solution

   An isosceles trapezoid has exactly one pair of parallel sides (bases) and the non-parallel sides (legs) are congruent. In coordinate geometry, parallel lines have equal slopes, and congruent segments have equal lengths calculated using the distance formula.

5. Sophia is proving that triangle ABC with vertices A(8, 11), B(14, 23), and C(20, 11) is isosceles using coordinate geometry. She wants to show that two sides have equal length. Which pair of sides should she compare to prove the triangle is isosceles? Answer: A. AB and AC Solution: Calculate length AB using distance formula: AB = sqrt((14-8)^2 + (23-11)^2) = sqrt(6^2 + 12^2) = sqrt(36 + 144) = sqrt(180) = 6√5 Calculate length BC using distance formula: BC = sqrt((20-14)^2 + (11-23)^2) = sqrt(6^2 + (-12)^2) = sqrt(36 + 144) = sqrt(180) = 6√5 Calculate length AC using…
   Full step-by-step solution

   Step 1: Calculate length AB using distance formula: AB = sqrt((14-8)^2 + (23-11)^2) = sqrt(6^2 + 12^2) = sqrt(36 + 144) = sqrt(180) = 6√5 Step 2: Calculate length BC using distance formula: BC = sqrt((20-14)^2 + (11-23)^2) = sqrt(6^2 + (-12)^2) = sqrt(36 + 144) = sqrt(180) = 6√5 Step 3: Calculate length AC using distance formula: AC = sqrt((20-8)^2 + (11-11)^2) = sqrt(12^2 + 0^2) = sqrt(144) = 12 Step 4: Compare the lengths: AB = 6√5, BC = 6√5, AC = 12 Step 5: Since AB = BC, the triangle is isosceles with AB and BC as the equal sides The correct answer is AB and AC.

6. Prove that quadrilateral Sophia with vertices A(1,1), B(6,4), C(11,1), and D(6,-2) is a rhombus using coordinate geometry. Answer: All sides are equal length (5 units) and diagonals are perpendicular (slopes 0 and undefined). Solution: A rhombus is a quadrilateral with all sides equal in length. Additionally, the diagonals of a rhombus are perpendicular bisectors of each other.
   Full step-by-step solution

   A rhombus is a quadrilateral with all sides equal in length. Additionally, the diagonals of a rhombus are perpendicular bisectors of each other. You can use the distance formula to verify equal side lengths and the slope formula to check if diagonals are perpendicular.

7. Isabella is proving that a triangular garden plot with vertices at (2, 7), (12, 17), and (22, 7) is isosceles. She needs to find the length of the base to calculate the area. What is the length of the base of this triangle? Answer: 20 Solution: Identify the vertices: A(2, 7), B(12, 17), C(22, 7) Calculate the distances between each pair of points using the distance formula: distance = sqrt((x2 - x1)^2 + (y2 - y1)^2) AB = sqrt((12 - 2)^2 + (17 - 7)^2) = sqrt(10^2 + 10^2) = sqrt(100 + 100) = sqrt(200) BC = sqrt((22 - 12)^2 + (7 - 17)^2)…
   Full step-by-step solution

   Step 1: Identify the vertices: A(2, 7), B(12, 17), C(22, 7) Step 2: Calculate the distances between each pair of points using the distance formula: distance = sqrt((x2 - x1)^2 + (y2 - y1)^2) Step 3: AB = sqrt((12 - 2)^2 + (17 - 7)^2) = sqrt(10^2 + 10^2) = sqrt(100 + 100) = sqrt(200) Step 4: BC = sqrt((22 - 12)^2 + (7 - 17)^2) = sqrt(10^2 + (-10)^2) = sqrt(100 + 100) = sqrt(200) Step 5: AC = sqrt((22 - 2)^2 + (7 - 7)^2) = sqrt(20^2 + 0^2) = sqrt(400) = 20 Step 6: Since AB = BC, the triangle is isosceles with AC as the base Step 7: The length of the base AC is 20

8. Isabella draws a quadrilateral on a coordinate plane with vertices at A(2,2), B(12,2), C(17,7), and D(7,7). She wants to prove this quadrilateral is a parallelogram using coordinate geometry. What is the length of side BC? Answer: 5√2 Solution: To find the length of a line segment between two points on a coordinate plane, we use the distance formula.
   Full step-by-step solution

   To find the length of a line segment between two points on a coordinate plane, we use the distance formula. This formula comes from the Pythagorean theorem and calculates the straight-line distance between any two points. When working with quadrilaterals, finding side lengths helps determine properties like whether opposite sides are equal, which is one way to prove a quadrilateral is a parallelogram.