Question

Do the points \(P (1, 0)\), \(Q (- 5, 0)\) and \(R (- 2, 5)\) form a triangle ? If so, name the type of triangle formed.

Hint:

If the points were collinear, the area of the triangle calculated by the coordinates would be zero.
In coordinate geometry, check for equal side lengths first to quickly identify Isosceles or Equilateral types.

Answer 1

Given Points:
\(P(1, 0)\), \(Q(-5, 0)\), \(R(-2, 5)\)

Step 1: Find the distances PQ, QR and RP

Distance PQ:
\[ PQ = \sqrt{(1 - (-5))^2 + (0 - 0)^2} = \sqrt{6^2} = 6 \]

Distance QR:
\[ QR = \sqrt{(-5 + 2)^2 + (0 - 5)^2} = \sqrt{(-3)^2 + (-5)^2} = \sqrt{9 + 25} = \sqrt{34} \]

Distance RP:
\[ RP = \sqrt{(1 + 2)^2 + (0 - 5)^2} = \sqrt{3^2 + (-5)^2} = \sqrt{9 + 25} = \sqrt{34} \]

Step 2: Check if triangle is formed
All three side lengths are non-zero and satisfy triangle inequalities.
So, the three points do form a triangle.

Step 3: Identify the type of triangle
We have:
PQ = 6 QR = √34 RP = √34 Since QR = RP, two sides are equal.

Final Answer:
Yes, the points form a triangle.
The triangle formed is an isosceles triangle.