Question : ABC is an isosceles right-angle triangle. $\angle ABC = 90 ^{\circ}$ and AB = 12 cm. What is the ratio of the radius of the circle inscribed in it to the radius of the circle circumscribing $\triangle ABC$?
Option 1: $6–\sqrt{2}: 3 \sqrt{2}$
Option 2: $2–\sqrt{2}: \sqrt{2}$
Option 3: $6–3 \sqrt{2}: 1 \sqrt{2}$
Option 4: $6–3 \sqrt{2}: 6 \sqrt{2}$
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Correct Answer: $2–\sqrt{2}: \sqrt{2}$
Solution : Given: ABC is an isosceles right-angle triangle. $\angle ABC = 90 ^{\circ}$ and AB = 12 cm. In $\triangle ABC$, $H^2=B^2+P^2$ (Using Pythagoras theorem). ⇒ $H^2=12^2+12^2$ ⇒ $H^2=144+144=288$ ⇒ $H=\sqrt{288}=12\sqrt2$ cm Inradius of inscribed circle of right angled triangle = $\frac{(P + B – H)}{2}$. Circumradius of the circumscribed circle of right angle triangle = $\frac{H}{2}$ where $P$, $B$, and $H$ are the perpendicular, base, and hypotenuse. Inradius of inscribed circle of right angled triangle = $\frac{(12 + 12 – 12\sqrt2)}{2}=6(2–\sqrt2)$. Circumradius of the circumscribed circle of the right angle triangle = $\frac{12\sqrt2}{2}=6\sqrt2$. The ratio of the radius of the circle inscribed in it to the radius of the circle circumscribing triangle ABC = $6(2–\sqrt2):6\sqrt2=2–\sqrt{2}: \sqrt{2}$. Hence, the correct answer is $2–\sqrt{2}: \sqrt{2}$.
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Question : ABC is an equilateral triangle with a side of 12 cm. What is the length of the radius of the circle inscribed in it?
Question : $AB$ and $AC$ are the two tangents to a circle whose radius is 6 cm. If $\angle BAC=60^{\circ}$, then what is the value (in cm) of $\sqrt{\left ( AB^{2}+AC^{2} \right )}?$
Question : ABC is an isosceles triangle inscribed in a circle. If AB = AC = $12\sqrt{5}$ cm and BC = 24 cm, then the radius of circle is:
Question : ABC is an equilateral triangle. If the area of the triangle is $36 \sqrt{3}$, then what is the radius of the circle circumscribing the $\triangle ABC$?
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