CAT 2024 Slot 2 · Quant · Question 14

Two of the vertices, \left( 1,2 \right) and \left( 7,2 \right), share the same y-coordinate, so that side is horizontal with length 7-1=6.

Two of them, \left( 1,2 \right) and \left( 1,10 \right), share the same x-coordinate, so that side is vertical with length 10-2=8.

These two sides meet at \left( 1,2 \right) at a right angle, so the triangle is right-angled there.

The hypotenuse joins \left( 7,2 \right) and \left( 1,10 \right), with length \sqrt{{6}^{2}+{8}^{2}}=\sqrt{100}=10.

For a right-angled triangle with legs a and b and hypotenuse h, the inradius is r=\frac{a+b-h}{2}.

Substituting, r=\frac{6+8-10}{2}=\frac{4}{2}=2.

The radius of the incircle is 2 units.

Shortcut. Legs 6 and 8 with hypotenuse 10 is the familiar 3-4-5 triangle doubled, so the hypotenuse needs no calculation. The right-angle inradius formula \frac{a+b-h}{2} then finishes it faster than the general r=\frac{\text{Area}}{s}.