Question

The triangle joining the points $P(2,7)Q(4,-1)$ and $R(-2,6)$ is

• A. Isosceles triangle
• B. Equilateral triangle
• C. Right-angled triangle
• D. None of this

Answer 1

The correct option is C

Right-angled triangle

Explanation for correct option

Given that, $P(2,7)Q(4,–1)$ and $R(-2,6)$

The distance between $PQ$ is

$$\begin{gathered} PQ=\sqrt{{\left(2-4\right)}^2+{\left(7+1\right)}^2} \\ =\sqrt{2^2+8^2} \\ =\sqrt{68} \end{gathered}$$

The distance between $PR$ is

$$\begin{gathered} PR=\sqrt{{\left(2+2\right)}^2+{\left(7-6\right)}^2} \\ =\sqrt{4^2+1} \\ =\sqrt{17} \end{gathered}$$

The distance between $QR$is

$$\begin{gathered} QR=\sqrt{{\left(4+2\right)}^2+{\left(-1-6\right)}^2} \\ =\sqrt{36+49} \\ =\sqrt{85} \end{gathered}$$

Therefore,

$P{Q}^2+P{R}^2$

$$\begin{gathered} ={\left(\sqrt{68}\right)}^2+{\left(\sqrt{17}\right)}^2 \\ =68+17 \\ =85 \\ =Q{R}^2 \\ \therefore P{Q}^2+P{R}^2=Q{R}^2 \end{gathered}$$

As the length of the sides of the triangle satisfy Pythagoras theorem sot the given points $P(2,7)Q(4,–1)$ and$R(-2,6)$ form a right angled triangle.

Hence, the correct option is $\mathrm{Option}\left(\mathrm{C}\right)$