Question

\[{s_1}\] and \[{s_2}\] are the inscribed and circumscribed circles of a triangle with sides 3, 4 and 5. Then (area of ${s_1}$) / (area of ${s_2}$)
A. \[16/25\]
B. \[4/25\]
C. \[9/25\]
D. \[9/16\]

Answer 1

Hint: In this question, we have been given that there are two circles \[{s_1}\] and \[{s_2}\] which are inscribed and circumscribed in a triangle. And we have to find the ratio of area of \[{s_1}\] to the ratio of the area of \[{s_2}\]. A thing to note is that the given triangle (with sides 3, 4 and 5) is a right-angled triangle with its sides forming the most famous and the smallest Pythagoras triplet of 3-4-5 with 3 and 4 as the base and perpendicular pair and with 5 as the hypotenuse of the triangle. To calculate the areas of the two triangles, we follow the formulae for calculating the area of the respective triangle, put in the required values, evaluate the area, and then just divide them to find their ratio, and we are going to have our answer.

Complete step-by-step answer:
We are going to use the formula for the area of inscribed circle and circumscribed circle for which we need to know the radii of the two circles, which is:
Radius of inscribed circle \[r = \dfrac{{{\rm{area\ of \triangle}}}}{{{\rm{semi-perimeter\ of \triangle}}}}\]
and radius of circumscribed circle \[R = \dfrac{{{\rm{length\ of\ hypotenuse}}}}{{{\rm{2}} \times {\rm{sine\ of\ central\ angle}}}}\]
Complete step by step answer:
In the given question, the triangle given to us is a right-angled triangle. Hence, its central angle is \[90^\circ \].
Now, we evaluate the radii of the circles:
Radius of inscribed circle \[r = \dfrac{{{\rm{area\ of \triangle}}}}{{{\rm{semi-perimeter\ of \triangle}}}} = \dfrac{{\dfrac{1}{2} \times 3 \times 4}}{{\dfrac{{3 + 4 + 5}}{2}}} = \dfrac{6}{6} = 1\]
and radius of circumscribed circle \[R = \dfrac{{{\rm{length\ of\ hypotenuse}}}}{{{\rm{2}} \times {\rm{sine\ of\ central\ angle}}}} = \dfrac{5}{{2 \times \sin 90^\circ }} = \dfrac{5}{2}\]
So, the ratio of their areas is \[\dfrac{{\pi {r^2}}}{{\pi {R^2}}} = \dfrac{{{1^2}}}{{{{\left( {\dfrac{5}{2}} \right)}^2}}} = \dfrac{1}{{\dfrac{{25}}{4}}} = \dfrac{4}{{25}}\]
Hence, the ratio of the areas of the inscribed and circumscribed circles of the given triangle is \[\dfrac{4}{{25}}\].

Thus, the correct option is B).

Note: So, we saw that in solving questions like these, we first write down what has been given to us. Then we write down what we need to find out. Then we write down the formula (or formulae) which connects the two things (the known and the unknown). Then we put in the values of the knowns into the formula and find the unknown. We also look for some special or some different thing in the question (like in this question we saw that the given triangle is a right-angled triangle) with which we can put in the value of something (in this question it was the central angle, as the central angle for a right-angled triangle is \[90^\circ \] which is a known fact but was not given in the question directly) which is not directly given. Then we just put in the values and calculate the answer.