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Coordinate Geometry Proofs

Grade 10 · Mathematics · Worksheet 3

  1. Aroha is a landscape architect designing a triangular meditation garden for a community center. She plots the garden on a coordinate grid with vertices at A(9, 12), B(21, 4), and C(15, 20). To ensure the garden is symmetrical, she needs to prove that the triangle is isosceles. Using coordinate geometry, calculate the lengths of all three sides and identify which two sides are equal. Express all lengths in simplest radical form. Answer: ______________
  2. Aroha is a landscape architect designing a new garden for a community center. She has plotted four key points on a coordinate grid to mark the corners of a new flower bed: P(9, 12), Q(21, 16), R(17, 28), and S(5, 24). She wants to prove that this quadrilateral PQRS is a parallelogram so she can order a perfectly fitting border. Using coordinate geometry, calculate the slopes of all four sides and show that opposite sides are parallel to verify the shape is a parallelogram. Answer: ______________
  3. Prove that quadrilateral PQRS with vertices P(2, 7), Q(12, 17), R(22, 7), and S(12, -3) is a rhombus using coordinate geometry. Answer: ______________
  4. Sophia is designing a new rectangular garden for a community center. She places three corners of the garden at coordinates A(9, 12), B(21, 18), and C(15, 30) on a city planning grid. She needs to determine the coordinates of the fourth corner D such that quadrilateral ABCD is a parallelogram. Using coordinate geometry, find the coordinates of point D. Answer: ______________
  5. Matiu is a land surveyor mapping a new conservation area. He plots four boundary markers on a coordinate grid: P(-3, 7), Q(11, 9), R(15, -3), and S(1, -5). Matiu needs to prove that these points form a parallelogram for his official report. Using coordinate geometry and the concept of midpoints, verify that quadrilateral PQRS is a parallelogram. State the coordinates of the intersection point of the diagonals as your final answer. Answer: ______________
  6. A right triangle is positioned on a coordinate plane with vertices at A(0,0), B(6,0), and C(0,8). Point D lies on the hypotenuse AC such that AD:DC = 1:3. Calculate the coordinates of point D using section formula. Answer: ______________
  7. √( (5 - (-3))² + (8 - 2)² ) = ? Answer: ______________
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Answer Key & Explanations

Coordinate Geometry Proofs · Grade 10 · Worksheet 3

  1. Aroha is a landscape architect designing a triangular meditation garden for a community center. She plots the garden on a coordinate grid with vertices at A(9, 12), B(21, 4), and C(15, 20). To ensure the garden is symmetrical, she needs to prove that the triangle is isosceles. Using coordinate geometry, calculate the lengths of all three sides and identify which two sides are equal. Express all lengths in simplest radical form. Answer: AB = sqrt(208), AC = sqrt(100) = 10, BC = sqrt(292). Since none are equal, the triangle is NOT isosceles. (Note: Re-checking — AB = sqrt((21-9)^2+(4-12)^2) = sqrt(144+64)=sqrt(208); AC = sqrt((15-9)^2+(20-12)^2)=sqrt(36+64)=sqrt(100)=10; BC = sqrt((15-21)^2+(20-4)^2)=sqrt(36+256)=sqrt(292). No two sides are equal.) Solution: Calculate AB using A(9,12) and B(21,4). AB = sqrt((21-9)^2 + (4-12)^2) = sqrt(12^2 + (-8)^2) = sqrt(144 + 64) = sqrt(208) = 4*sqrt(13). Calculate AC using A(9,12) and C(15,20).
    Full step-by-step solution

    Step 1: Calculate AB using A(9,12) and B(21,4). AB = sqrt((21-9)^2 + (4-12)^2) = sqrt(12^2 + (-8)^2) = sqrt(144 + 64) = sqrt(208) = 4*sqrt(13). Step 2: Calculate AC using A(9,12) and C(15,20). AC = sqrt((15-9)^2 + (20-12)^2) = sqrt(6^2 + 8^2) = sqrt(36 + 64) = sqrt(100) = 10. Step 3: Calculate BC using B(21,4) and C(15,20). BC = sqrt((15-21)^2 + (20-4)^2) = sqrt((-6)^2 + 16^2) = sqrt(36 + 256) = sqrt(292) = 2*sqrt(73). Step 4: Compare lengths: AB = sqrt(208), AC = 10, BC = sqrt(292). No two sides are equal. Therefore, triangle ABC is NOT isosceles.

  2. Aroha is a landscape architect designing a new garden for a community center. She has plotted four key points on a coordinate grid to mark the corners of a new flower bed: P(9, 12), Q(21, 16), R(17, 28), and S(5, 24). She wants to prove that this quadrilateral PQRS is a parallelogram so she can order a perfectly fitting border. Using coordinate geometry, calculate the slopes of all four sides and show that opposite sides are parallel to verify the shape is a parallelogram. Answer: PQ and RS both have slope 1/3, QR and SP both have slope -3/2, so opposite sides are parallel, proving PQRS is a parallelogram. Solution: Calculate slope of PQ using points P(9, 12) and Q(21, 16). Slope = (16 - 12) / (21 - 9) = 4 / 12 = 1/3 Calculate slope of RS using points R(17, 28) and S(5, 24).
    Full step-by-step solution

    Step 1: Calculate slope of PQ using points P(9, 12) and Q(21, 16). Slope = (16 - 12) / (21 - 9) = 4 / 12 = 1/3 Step 2: Calculate slope of RS using points R(17, 28) and S(5, 24). Slope = (24 - 28) / (5 - 17) = (-4) / (-12) = 1/3 Since slope PQ = slope RS = 1/3, side PQ is parallel to side RS. Step 3: Calculate slope of QR using points Q(21, 16) and R(17, 28). Slope = (28 - 16) / (17 - 21) = 12 / (-4) = -3/2 Step 4: Calculate slope of SP using points S(5, 24) and P(9, 12). Slope = (12 - 24) / (9 - 5) = (-12) / 4 = -3/2 Since slope QR = slope SP = -3/2, side QR is parallel to side SP. Step 5: Both pairs of opposite sides are parallel (PQ ∥ RS and QR ∥ SP), therefore quadrilateral PQRS is a parallelogram. The answer is: PQ and RS both have slope 1/3, QR and SP both have slope -3/2, so opposite sides are parallel, proving PQRS is a parallelogram.

  3. Prove that quadrilateral PQRS with vertices P(2, 7), Q(12, 17), R(22, 7), and S(12, -3) is a rhombus using coordinate geometry. Answer: PQRS is a rhombus Solution: Calculate the length of side PQ using the distance formula: sqrt((12 - 2)^2 + (17 - 7)^2) = sqrt(10^2 + 10^2) = sqrt(100 + 100) = sqrt(200) = 10*sqrt(2).
    Full step-by-step solution

    Step 1: Calculate the length of side PQ using the distance formula: sqrt((12 - 2)^2 + (17 - 7)^2) = sqrt(10^2 + 10^2) = sqrt(100 + 100) = sqrt(200) = 10*sqrt(2). Step 2: Calculate the length of side QR: sqrt((22 - 12)^2 + (7 - 17)^2) = sqrt(10^2 + (-10)^2) = sqrt(100 + 100) = sqrt(200) = 10*sqrt(2). Step 3: Calculate the length of side RS: sqrt((12 - 22)^2 + (-3 - 7)^2) = sqrt((-10)^2 + (-10)^2) = sqrt(100 + 100) = sqrt(200) = 10*sqrt(2). Step 4: Calculate the length of side SP: sqrt((2 - 12)^2 + (7 - (-3))^2) = sqrt((-10)^2 + 10^2) = sqrt(100 + 100) = sqrt(200) = 10*sqrt(2). Step 5: All four sides are equal to 10*sqrt(2), so quadrilateral PQRS is a rhombus. The answer is PQRS is a rhombus.

  4. Sophia is designing a new rectangular garden for a community center. She places three corners of the garden at coordinates A(9, 12), B(21, 18), and C(15, 30) on a city planning grid. She needs to determine the coordinates of the fourth corner D such that quadrilateral ABCD is a parallelogram. Using coordinate geometry, find the coordinates of point D. Answer: D(3, 24) Solution: In a parallelogram, the diagonals bisect each other. For quadrilateral ABCD, the diagonals are AC and BD. Midpoint of AC = ((9 + 15)/2, (12 + 30)/2) = (24/2, 42/2) = (12, 21) Let D = (x, y).
    Full step-by-step solution

    Step 1: In a parallelogram, the diagonals bisect each other. For quadrilateral ABCD, the diagonals are AC and BD. The midpoint of AC equals the midpoint of BD. Step 2: Find the midpoint of AC. Midpoint of AC = ((9 + 15)/2, (12 + 30)/2) = (24/2, 42/2) = (12, 21) Step 3: Let D = (x, y). The midpoint of BD is ((21 + x)/2, (18 + y)/2). Step 4: Set the midpoints equal. (21 + x)/2 = 12 and (18 + y)/2 = 21 Step 5: Solve for x. 21 + x = 24 x = 3 Step 6: Solve for y. 18 + y = 42 y = 24 Step 7: Therefore, the coordinates of D are (3, 24). Check: Verify that opposite sides are parallel using slopes. Slope of AB = (18 - 12)/(21 - 9) = 6/12 = 1/2 Slope of DC = (30 - 24)/(15 - 3) = 6/12 = 1/2 Slope of BC = (30 - 18)/(15 - 21) = 12/(-6) = -2 Slope of AD = (24 - 12)/(3 - 9) = 12/(-6) = -2 Opposite sides are parallel, confirming D(3, 24) is correct. The answer is D(3, 24).

  5. Matiu is a land surveyor mapping a new conservation area. He plots four boundary markers on a coordinate grid: P(-3, 7), Q(11, 9), R(15, -3), and S(1, -5). Matiu needs to prove that these points form a parallelogram for his official report. Using coordinate geometry and the concept of midpoints, verify that quadrilateral PQRS is a parallelogram. State the coordinates of the intersection point of the diagonals as your final answer. Answer: (6, 2) Solution: Identify the diagonals of quadrilateral PQRS. The diagonals are PR and QS. Calculate the midpoint of diagonal PR using the midpoint formula: ((x1 + x2)/2, (y1 + y2)/2).
    Full step-by-step solution

    Step 1: Identify the diagonals of quadrilateral PQRS. The diagonals are PR and QS. Step 2: Calculate the midpoint of diagonal PR using the midpoint formula: ((x1 + x2)/2, (y1 + y2)/2). Coordinates of P: (-3, 7) Coordinates of R: (15, -3) Midpoint of PR = ((-3 + 15)/2, (7 + (-3))/2) = (12/2, 4/2) = (6, 2) Step 3: Calculate the midpoint of diagonal QS. Coordinates of Q: (11, 9) Coordinates of S: (1, -5) Midpoint of QS = ((11 + 1)/2, (9 + (-5))/2) = (12/2, 4/2) = (6, 2) Step 4: Compare the two midpoints. Midpoint of PR = (6, 2) Midpoint of QS = (6, 2) Since the midpoints are the same, the diagonals bisect each other. Step 5: Conclusion. Because the diagonals of quadrilateral PQRS bisect each other, PQRS is a parallelogram. The intersection point of the diagonals is (6, 2). The answer is (6, 2).

  6. A right triangle is positioned on a coordinate plane with vertices at A(0,0), B(6,0), and C(0,8). Point D lies on the hypotenuse AC such that AD:DC = 1:3. Calculate the coordinates of point D using section formula. Answer: (1.5, 2) Solution: The section formula is used to find coordinates of a point dividing a line segment joining two given points in a specific ratio.
    Full step-by-step solution

    The section formula is used to find coordinates of a point dividing a line segment joining two given points in a specific ratio. This concept connects coordinate geometry with ratio and proportion principles, allowing precise location of points along line segments in geometric configurations.

  7. √( (5 - (-3))² + (8 - 2)² ) = ? Answer: 10 Solution: We are given: √( (5 - (-3))² + (8 - 2)² ) Simplify inside the parentheses. First part: 5 - (-3) = 5 + 3 = 8 Second part: 8 - 2 = 6 So the expression becomes: √( (8)² + (6)² ) Square each term.
    Full step-by-step solution

    We are given: √( (5 - (-3))² + (8 - 2)² ) Step 1: Simplify inside the parentheses. First part: 5 - (-3) = 5 + 3 = 8 Second part: 8 - 2 = 6 So the expression becomes: √( (8)² + (6)² ) Step 2: Square each term. (8)² = 64 (6)² = 36 So we have: √(64 + 36) Step 3: Add inside the square root. 64 + 36 = 100 So we have: √(100) Step 4: Take the square root. √(100) = 10 Final answer: 10