Question

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

Options

• equilateral

• isosceles

• scalene

• right angled

Answer 1

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 ≠ QR, two sides are equal → triangle is isosceles.

Slopes: slope `PQ = (3 - 6)/(-5 - 0) = 3/5`

Slope `PR = (1 - 6)/(3 - 0) = (-5)/3`

Their product = `(3/5)(-5/3) = -1`, so PQ ⟂ PR → a right angle at P. 

Therefore, the triangle is an isosceles right triangle.