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

Step 1: Understanding the Concept:
To determine if three points form a triangle, the sum of the lengths of any two sides must be greater than the length of the third side.
If a triangle is formed, we can identify its type (Scalene, Isosceles, or Equilateral) based on the lengths of its sides.
Step 2: Key Formula or Approach:
Distance formula: \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)
Step 3: Detailed Explanation:
Let the points be \(P(1, 0)\), \(Q(-5, 0)\), and \(R(-2, 5)\).
Calculate the length of side \(PQ\):
\[ PQ = \sqrt{(-5 - 1)^2 + (0 - 0)^2} = \sqrt{(-6)^2 + 0} = 6 \]
Calculate the length of side \(QR\):
\[ QR = \sqrt{(-2 - (-5))^2 + (5 - 0)^2} = \sqrt{(3)^2 + (5)^2} = \sqrt{9 + 25} = \sqrt{34} \approx 5.83 \]
Calculate the length of side \(RP\):
\[ RP = \sqrt{(1 - (-2))^2 + (0 - 5)^2} = \sqrt{(3)^2 + (-5)^2} = \sqrt{9 + 25} = \sqrt{34} \approx 5.83 \]
Checking the triangle inequality:
\(PQ + QR = 6 + 5.83 = 11.83>5.83\) (\(RP\))
\(QR + RP = 5.83 + 5.83 = 11.66>6\) (\(PQ\))
\(RP + PQ = 5.83 + 6 = 11.83>5.83\) (\(QR\))
Since the sum of any two sides is greater than the third side, the points form a triangle.
Since \(QR = RP = \sqrt{34}\), two sides are equal in length.
Step 4: Final Answer:
The points form an Isosceles triangle.