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.
ScanSolve AI Answer
Here are the steps to solve the problem: Part 1: Find the length of the sides of the triangle Step 1: Use the distance formula D = sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2) to find the length of each side. Length of PQ: For P(3, 2) and Q(-1, 5) PQ = sqrt((-1 - 3)^2 + (5 - 2)^2) = sqrt((-4)^2 + (3)^2) = sqrt(16 + 9) = sqrt(25) = 5 Length of QR: For Q(-1, 5) and R(2, 1) QR = sqrt((2 - (-1))^2 + (1 - 5)^2) = sqrt((3)^2 + (-4)^2) = sqrt(9 + 16) = sqrt(25) = 5 Length of RP: For R(2, 1) and P(3, 2) RP = sqrt((3 - 2)^2 + (2 - 1)^2) = sqrt((1)^2 + (1)^2) = sqrt(1 + 1) = sqrt(2) The lengths of the sides are: PQ = 5 QR = 5 RP = sqrt(2) Part 2: Use the dot product to find an angle of the triangle Step 1: Choose an angle to find. Let's find the angle at vertex P, QPR. First, determine the vectors PQ and PR. PQ = Q - P = (-1 - 3, 5 - 2) = -4 \\ 3 PR = R - P = (2 - 3, 1 - 2) = -1 \\ -1 Step 2: Calculate the dot product of PQ and PR. PQ · PR = (-4)(-1) + (3)(-1) = 4 - 3 = 1 Step 3: Calculate the magnitudes of PQ and PR. |PQ| = sqrt((-4)^2 + (3)^2) = sqrt(16 + 9) = sqrt(25) = 5 |PR| = sqrt((-1)^2 + (-1)^2) = sqrt(1 + 1) = sqrt(2) Step 4: Use the dot product formula = a · b|a| |b| to find the angle . ( QPR) = PQ · PR|PQ| |PR| = (1)/(5 · sqrt(2)) = (1)/(5sqrt(2)) Rationalize the denominator: ( QPR) = (1)/(5sqrt(2)) × sqrt(2)sqrt(2) = sqrt(2)10 Step 5: Calculate the angle. QPR = (sqrt(2)10) ≈ (0.14142) ≈ 81.87^ An angle of the triangle is (sqrt(2)10) ≈ 81.87^. Send me the next one 📸