Question

Find the type of triangle formed by the points P(-5, 6), Q(-4, -2) and R(7, 5).

Answer 1

Hint: To find the type of triangle, we need to find the length of the sides of the triangle, and then categorize it into equilateral, isosceles, scalene or a right-angled triangle. If all three sides are equal, then the triangle is equilateral. If two are equal, then it is isosceles, and if none of them are equal, it is scalene. If the sum of squares of two sides is equal to the third side, then the triangle is said to be right-angled. The distance formula will be used to find the distance between the points as-
${\text{D}} = \sqrt {{{\left( {{{\text{x}}_2} - {{\text{x}}_1}} \right)}^2} + {{\left( {{{\text{y}}_2} - {{\text{y}}_1}} \right)}^2}} $

Complete step-by-step answer:

We will now use the distance formula to find the length of the three sides of the triangle. It is given by-
$\begin{align}
  &PQ = \sqrt {{{\left( { - 4 - \left( { - 5} \right)} \right)}^2} + {{\left( { - 2 - 6} \right)}^2}} = \sqrt {{1^2} + {{\left( { - 8} \right)}^2}} = \sqrt {1 + 64} = \sqrt {65} \\
  &QR = \sqrt {{{\left( {7 - \left( { - 4} \right)} \right)}^2} + {{\left( {5 - \left( { - 2} \right)} \right)}^2}} = \sqrt {{{11}^2} + {7^2}} = \sqrt {121 + 49} = \sqrt {170} \\
  &PR = \sqrt {{{\left( {7 - \left( { - 5} \right)} \right)}^2} + {{\left( {5 - 6} \right)}^2}} = \sqrt {{{12}^2} + {{\left( { - 1} \right)}^2}} = \sqrt {144 + 1} = \sqrt {145} \\
\end{align} $

From these distances, we can clearly see that none of the sides are equal in length. Now we will apply the Pythagoras Theorem to check if the triangle is right-angled or not.

The Pythagoras Theorem is given as-
${{\text{a}}^2} + {{\text{b}}^2} = {{\text{c}}^2}$
Here, c is the longest side.

In $\vartriangle PQR$ we can check the Pythagoras Theorem as-
$\begin{align}
  &P{Q^2} + P{R^2} = Q{R^2}\left( \because\text{ QR is the longest} \right) \\
  &{\left( {\sqrt {65} } \right)^2} + {\left( {\sqrt {145} } \right)^2} = {\left( {\sqrt {170} } \right)^2} \\
  &65 + 145 = 170 \\
  &210 \ne 170 \\
\end{align} $
We can clearly see that this triangle does not obey Pythagoras Theorem as well, hence the given triangle PQR is a scalene triangle.
This is the required answer.

Note: In such types of questions, we need to know multiple concepts related to coordinate geometry and triangles. To check for a right-angled triangle, we can also find the slopes of the sides of the triangle and check if they are at a right angle to each other. However, this is a lengthy method and pythagoras theorem should be used.