**Question:**

Name the type of triangle formed by the points A (-5, 6), B(- 4, – 2) and C(7, 5).

**Solution:**

To find the type of triangle, first we determine the length of all three sides and see whatever condition of triangle is satisfy by these sides.

Now, using distance formula between two points,

$A B=\sqrt{(-4+5)^{2}+(-2-6)^{2}}$

$=\sqrt{(1)^{2}+(-8)^{2}}$

$=\sqrt{1+64}=\sqrt{65} \quad\left[\because d=\sqrt{\left(x_{2}-x_{1}\right)+\left(y_{2}-y_{1}\right)^{2}}\right]$

$B C=\sqrt{(7+4)^{2}+(5+2)^{2}}=\sqrt{(11)^{2}+(7)^{2}}$

$=\sqrt{121+49}=\sqrt{170}$

and

$C A=\sqrt{(-5-7)^{2}+(6-5)^{2}}=\sqrt{(-12)^{2}+(1)^{2}}$

$=\sqrt{144+1}=\sqrt{145}$

We see that, AB ≠ BC ≠ CA

and not hold the condition of Pythagoras in a ΔABC.

i.e., (Hypotenuse)2 =( Base)2 + (Perpendicular)2

Hence, the required triangle is scalene because all of its sides are not equal i.e., different to each other.