The points P(0, 6), Q(–5, 3) and R(3, 1) are the vertices of a triangle, which is ______.

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The points P(0, 6), Q(–5, 3) and R(3, 1) are the vertices of a triangle, which is right angled. Explanation: Using the distance formula. Distance between (x1, y1) and (x2, y2) = `sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2)`. PQ = Distance between P(0, 6) and Q(–5, 3) = sqrt((-5 - 0)^2 + (3 - 6)^2) = sqrt(25 + 9) = sqrt(34) PR = Distance between P(0, 6) and R(3, 1) = sqrt((3 - 0)^2 + (1 - 6)^2) = sqrt(9 + 25) = sqrt(34) QR = Distance between Q(–5, 3) and R(3, 1) = sqrt((3 + 5)^2 + (1 - 3)^2) = sqrt(64 + 4) = sqrt(68) Since PQ = PR $\neq$ QR, two sides are equal $\to$ triangle is isosceles. Slopes: slope PQ = (3 - 6)/(-5 - 0) = $\frac{3}{5}$ Slope `PR = (1 - 6)/(3 - 0) = (-5)/3 Their product = ($\frac{3}{5}$)(-$\frac{5}{3}$) = -1`, so PQ ⟂ PR $\to$ a right angle at P. Therefore, the triangle is an isosceles right triangle.

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