Coordinate Proofs

lessonbunny.com

Answer Key & Explanations

Coordinate Proofs · Grade 10 · Worksheet 3

1. Prove that triangle Aroha with vertices A(9,0), B(15,0), C(12,4) is isosceles using coordinate geometry. Answer: Triangle Aroha is isosceles because AB = 6, AC = 5, and BC = 5, showing that two sides (AC and BC) are equal in length. Solution: An isosceles triangle has at least two sides of equal length. The distance formula can be used to find the length between any two points in the coordinate plane.
   Full step-by-step solution

   An isosceles triangle has at least two sides of equal length. The distance formula can be used to find the length between any two points in the coordinate plane. If two of the calculated distances are equal, then the triangle is isosceles.

2. Prove that triangle ABC with vertices A(0,0), B(10,0), and C(5,5) is isosceles using coordinate geometry. Answer: AB = 10, AC = √50, BC = √50, AC = BC Solution: Calculate AB using distance formula: AB = √[(10-0)² + (0-0)²] = √[100 + 0] = √100 = 10 Calculate AC using distance formula: AC = √[(5-0)² + (5-0)²] = √[25 + 25] = √50 Calculate BC using distance formula: BC = √[(5-10)² + (5-0)²] = √[25 + 25] = √50 Compare side lengths: AB = 10, AC = √50, BC =…
   Full step-by-step solution

   Step 1: Calculate AB using distance formula: AB = √[(10-0)² + (0-0)²] = √[100 + 0] = √100 = 10 Step 2: Calculate AC using distance formula: AC = √[(5-0)² + (5-0)²] = √[25 + 25] = √50 Step 3: Calculate BC using distance formula: BC = √[(5-10)² + (5-0)²] = √[25 + 25] = √50 Step 4: Compare side lengths: AB = 10, AC = √50, BC = √50 Step 5: Since AC = BC = √50, the triangle has two equal sides, proving it is isosceles.

3. Prove that quadrilateral Hana with vertices H(2,4), A(6,8), N(10,4), and A(6,0) is a square. Answer: The quadrilateral is a square because all four sides are equal in length (distance = sqrt(32)) and all angles are right angles (adjacent sides have slopes that are negative reciprocals). Solution: A square is a special quadrilateral where all four sides are equal and all four angles are right angles.
   Full step-by-step solution

   A square is a special quadrilateral where all four sides are equal and all four angles are right angles. In coordinate geometry, we can use the distance formula to verify equal side lengths and the slope formula to verify perpendicular sides (slopes are negative reciprocals).

4. Prove that triangle Emma with vertices E(1,1), M(5,1), M(3,5) is isosceles using coordinate geometry. Answer: The triangle is isosceles because EM = MM = 4 units Solution: An isosceles triangle has at least two sides of equal length. The distance formula helps calculate side lengths using coordinates. If two sides have the same length, the triangle is isosceles.
5. Isabella is designing a garden with a triangular flower bed. She places markers at coordinates A(8, 12), B(16, 24), and C(24, 12). She wants to prove that this triangle is isosceles. Using coordinate geometry, which of the following methods would correctly prove that triangle ABC is isosceles? Answer: B. Show that two sides have equal lengths using the distance formula Solution: To prove a triangle is isosceles using coordinate geometry, you need to demonstrate that two sides have equal length. The distance formula calculates side lengths from coordinates.
   Full step-by-step solution

   To prove a triangle is isosceles using coordinate geometry, you need to demonstrate that two sides have equal length. The distance formula calculates side lengths from coordinates. Equal slopes would indicate parallel lines, not equal sides. A right angle would make it a right triangle, not necessarily isosceles. Different side lengths would prove it's scalene, not isosceles.

6. Isabella is designing a triangular banner for her school's science fair. She places the vertices at coordinates A(0,0), B(8,0), and C(4,12). She wants to prove that the banner forms an isosceles triangle. Which of the following methods would correctly prove this? Answer: C. Show that two sides have equal lengths Solution: An isosceles triangle is defined by having at least two sides of equal length. While slopes can indicate parallel lines and right angles can identify right triangles, neither directly proves a triangle is isosceles.
   Full step-by-step solution

   An isosceles triangle is defined by having at least two sides of equal length. While slopes can indicate parallel lines and right angles can identify right triangles, neither directly proves a triangle is isosceles. The most direct method is to calculate the distances between vertices using the distance formula and confirm that two sides are equal in length. For example, with different coordinates like D(0,0), E(6,0), F(3,8), you would calculate DE, EF, and DF to see if any two match.

7. Prove that quadrilateral Sophia with vertices A(1,1), B(6,1), C(6,6), D(1,6) is a square using coordinate geometry. Answer: A. All sides equal and diagonals equal Solution: AB = sqrt((6-1)^2 + (1-1)^2) = sqrt(25 + 0) = 5 BC = sqrt((6-6)^2 + (6-1)^2) = sqrt(0 + 25) = 5 CD = sqrt((1-6)^2 + (6-6)^2) = sqrt(25 + 0) = 5 DA = sqrt((1-1)^2 + (1-6)^2) = sqrt(0 + 25) = 5 All sides equal 5 Check if adjacent sides are perpendicular using slope formula Slope AB = (1-1)/(6-1) =…
   Full step-by-step solution

   Step 1: Calculate side lengths using distance formula AB = sqrt((6-1)^2 + (1-1)^2) = sqrt(25 + 0) = 5 BC = sqrt((6-6)^2 + (6-1)^2) = sqrt(0 + 25) = 5 CD = sqrt((1-6)^2 + (6-6)^2) = sqrt(25 + 0) = 5 DA = sqrt((1-1)^2 + (1-6)^2) = sqrt(0 + 25) = 5 All sides equal 5 Step 2: Check if adjacent sides are perpendicular using slope formula Slope AB = (1-1)/(6-1) = 0/5 = 0 Slope BC = (6-1)/(6-6) = 5/0 = undefined (vertical line) Since horizontal and vertical lines are perpendicular, AB ⟂ BC Step 3: Check diagonals using distance formula AC = sqrt((6-1)^2 + (6-1)^2) = sqrt(25 + 25) = sqrt(50) = 5√2 BD = sqrt((1-6)^2 + (6-1)^2) = sqrt(25 + 25) = sqrt(50) = 5√2 Diagonals are equal Step 4: Conclusion All sides equal, adjacent sides perpendicular, and diagonals equal - this proves quadrilateral Sophia is a square. The correct answer is All sides equal and diagonals equal.