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  4. In the right angle triangle as shown, an altitude is drawn from the right angle vertex to the hypotenuse. Circles are inscribed within each of the smaller triangles. What is the distance between the centres of these circles? <img class=img-fluid question-image alt=Question src=https://cdn.tardigrade.in/q/nta/m-6qf2kxvqmao8yvlr.jpg />

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Q. In the right angle triangle as shown, an altitude is drawn from the right angle vertex to the hypotenuse. Circles are inscribed within
each of the smaller triangles. What is the distance between the centres of these circles?
Question

NTA AbhyasNTA Abhyas 2022

Solution:

Let the $\Delta $ be
$AC=\sqrt{15^{2} + 20^{2}}=25$
$BD=12$
$AD=9$
$CD=16$
Solution
$r_{1}=\frac{\Delta }{S}=\frac{\frac{\frac{1}{2} \times 12 \times 9}{12 + 9 + 15}}{2}\Rightarrow \frac{9 \times 12}{36}=3$
$r_{2}=\frac{\Delta }{S}=\frac{\frac{\frac{1}{2} \times 16 \times 12}{20 + 16 + 12}}{2}\Rightarrow \frac{16 \times 12}{48}=4$
distance $C_{1}C_{2}$
$=\sqrt{\left(r_{2} - r_{1}\right)^{2} + \left(r_{1} + r_{2}\right)^{2}}=\sqrt{1 + 49}=\sqrt{50}$