P(3,2) Q(-1,5) and R(2,1) are vertices of a triangle. (i) Find the length of sides of the triangle. (ii) Use the dot product to find angles of the triangle.

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Asked on July 10, 2026Mathematics

This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.

Accepted Answer · 2026-07-10T20:33:36.806Z

Given the vertices of the triangle P(3,2), Q(-1,5), and R(2,1). i) Find the length of sides of the triangle. Step 1: Calculate the length of side PQ using the distance formula d = sqrt((x_2-x_1)^2 + (y_2-y_1)^2). PQ = sqrt((-1-3)^2 + (5-2)^2) = sqrt((-4)^2 + (3)^2) = sqrt(16 + 9) = sqrt(25) = 5 Step 2: Calculate the length of side QR. QR = sqrt((2-(-1))^2 + (1-5)^2) = sqrt((3)^2 + (-4)^2) = sqrt(9 + 16) = sqrt(25) = 5 Step 3: Calculate the length of side RP. RP = sqrt((3-2)^2 + (2-1)^2) = sqrt((1)^2 + (1)^2) = sqrt(1 + 1) = sqrt(2) ≈ 1.414 The lengths of the sides are PQ=5, QR=5, RP=sqrt(2) ≈ 1.414. ii) Use the dot product to find angles of the triangle. Step 4: Find the angle at P (angle QPR). First, find the vectors PQ and PR. PQ = Q - P = (-1-3, 5-2) = (-4, 3) PR = R - P = (2-3, 1-2) = (-1, -1) Now, calculate their dot product and magnitudes: PQ · PR = (-4)(-1) + (3)(-1) = 4 - 3 = 1 |PQ| = 5 |PR| = sqrt((-1)^2 + (-1)^2) = sqrt(1+1) = sqrt(2) Using the formula = A · B|A| |B|: P = (1)/(5 · sqrt(2)) = (1)/(5sqrt(2)) P = ((1)/(5sqrt(2))) ≈ 81.87^ Step 5: Find the angle at Q (angle PQR). First, find the vectors QP and QR. QP = P - Q = (3-(-1), 2-5) = (4, -3) QR = R - Q = (2-(-1), 1-5) = (3, -4) Now, calculate their dot product and magnitudes: QP · QR = (4)(3) + (-3)(-4) = 12 + 12 = 24 |QP| = sqrt(4^2 + (-3)^2) = sqrt(16+9) = sqrt(25) = 5 |QR| = sqrt(3^2 + (-4)^2) = sqrt(9+16) = sqrt(25) = 5 Using the formula = A · B|A| |B|: Q = (24)/(5 · 5) = (24)/(25) Q = ((24)/(25)) ≈ 16.26^ Step 6: Find the angle at R (angle PRQ). First, find the vectors RP and RQ. RP = P - R = (3-2, 2-1) = (1, 1) RQ = Q - R = (-1-2, 5-1) = (-3, 4) Now, calculate their dot product and magnitudes: RP · RQ = (1)(-3) + (1)(4) = -3 + 4 = 1 |RP| = sqrt(1^2 + 1^2) = sqrt(2) |RQ| = sqrt((-3)^2 + 4^2) = sqrt(9+16) = sqrt(25) = 5 Using the formula = A · B|A| |B|: R = (1)/(sqrt(2) · 5) = (1)/(5sqrt(2)) R = ((1)/(5sqrt(2))) ≈ 81.87^ The angles of the triangle are P ≈ 81.87^, Q ≈ 16.26^, R ≈ 81.87^. 3 done, 2 left today. You're making progress.

Accepted Answer · 2026-07-10T20:33:36.806Z

Given the vertices of the triangle P(3,2), Q(-1,5), and R(2,1). i) Find the length of sides of the triangle. Step 1: Calculate the length of side PQ using the distance formula d = sqrt((x_2-x_1)^2 + (y_2-y_1)^2). PQ = sqrt((-1-3)^2 + (5-2)^2) = sqrt((-4)^2 + (3)^2) = sqrt(16 + 9) = sqrt(25) = 5 Step 2: Calculate the length of side QR. QR = sqrt((2-(-1))^2 + (1-5)^2) = sqrt((3)^2 + (-4)^2) = sqrt(9 + 16) = sqrt(25) = 5 Step 3: Calculate the length of side RP. RP = sqrt((3-2)^2 + (2-1)^2) = sqrt((1)^2 + (1)^2) = sqrt(1 + 1) = sqrt(2) ≈ 1.414 The lengths of the sides are PQ=5, QR=5, RP=sqrt(2) ≈ 1.414. ii) Use the dot product to find angles of the triangle. Step 4: Find the angle at P (angle QPR). First, find the vectors PQ and PR. PQ = Q - P = (-1-3, 5-2) = (-4, 3) PR = R - P = (2-3, 1-2) = (-1, -1) Now, calculate their dot product and magnitudes: PQ · PR = (-4)(-1) + (3)(-1) = 4 - 3 = 1 |PQ| = 5 |PR| = sqrt((-1)^2 + (-1)^2) = sqrt(1+1) = sqrt(2) Using the formula = A · B|A| |B|: P = (1)/(5 · sqrt(2)) = (1)/(5sqrt(2)) P = ((1)/(5sqrt(2))) ≈ 81.87^ Step 5: Find the angle at Q (angle PQR). First, find the vectors QP and QR. QP = P - Q = (3-(-1), 2-5) = (4, -3) QR = R - Q = (2-(-1), 1-5) = (3, -4) Now, calculate their dot product and magnitudes: QP · QR = (4)(3) + (-3)(-4) = 12 + 12 = 24 |QP| = sqrt(4^2 + (-3)^2) = sqrt(16+9) = sqrt(25) = 5 |QR| = sqrt(3^2 + (-4)^2) = sqrt(9+16) = sqrt(25) = 5 Using the formula = A · B|A| |B|: Q = (24)/(5 · 5) = (24)/(25) Q = ((24)/(25)) ≈ 16.26^ Step 6: Find the angle at R (angle PRQ). First, find the vectors RP and RQ. RP = P - R = (3-2, 2-1) = (1, 1) RQ = Q - R = (-1-2, 5-1) = (-3, 4) Now, calculate their dot product and magnitudes: RP · RQ = (1)(-3) + (1)(4) = -3 + 4 = 1 |RP| = sqrt(1^2 + 1^2) = sqrt(2) |RQ| = sqrt((-3)^2 + 4^2) = sqrt(9+16) = sqrt(25) = 5 Using the formula = A · B|A| |B|: R = (1)/(sqrt(2) · 5) = (1)/(5sqrt(2)) R = ((1)/(5sqrt(2))) ≈ 81.87^ The angles of the triangle are P ≈ 81.87^, Q ≈ 16.26^, R ≈ 81.87^. 3 done, 2 left today. You're making progress.

P(3,2) Q(-1,5) and R(2,1) are vertices of a triangle. (i) Find the length of sides of the triangle. (ii) Use the dot product to find angles of the triangle.

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P(3,2) Q(-1,5) and R(2,1) are vertices of a triangle. (i) Find the length of sides of the triangle. (ii) Use the dot product to find angles of the triangle.

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