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

To determine if the points P(0, 6), Q(-5, 3), and R(3, 1) are the vertices of a triangle, we can use the distance formula and the slope formula.

Step 1: Calculate the distances between the points. The distance between two points (x1, y1) and (x2, y2) is given by the formula: d = √((x2 - x1)^2 + (y2 - y1)^2)

Using this formula, we can calculate the distances PQ, QR, and RP.

Distance PQ: PQ = √((-5 - 0)^2 + (3 - 6)^2) = √((-5)^2 + (-3)^2) = √(25 + 9) = √34

Distance QR: QR = √((3 - (-5))^2 + (1 - 3)^2) = √((3 + 5)^2 + (-2)^2) = √(8^2 + 4) = √(64 + 4) = √68 = 2√17

Distance RP: RP = √((3 - 0)^2 + (1 - 6)^2) = √(3^2 + (-5)^2) = √(9 + 25) = √34

Step 2: Check if the sum of any two sides is greater than the third side. For a triangle to exist, the sum of any two sides must be greater than the third side.

Let's check if this condition is satisfied for the distances PQ, QR, and RP.

PQ + QR = √34 + 2√17 = √34 + √(2 * 17) = √34 + √34 = 2√34

QR + RP = 2√17 + √34

RP + PQ = √34 + √34 = 2√34

Since the sum of any two sides is greater than the third side, we can conclude that the points P(0, 6), Q(-5, 3), and R(3, 1) are the vertices of a triangle.