In right triangle ABC angle A =90 AB = 12cm and BC =20cm R is the way - Maths - Areas Related to Circles

Question

In right triangle ABC angle A =90° AB = 12cm and BC =20cm R is
the way incentre of triangle ABC Find the area of the circle drawn
with R as centre touching the sides of the triangle

Pooja Khanna · Accepted Answer

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"https://s3mn%20mnimgs%20com/img/shared/ck-files/ck_55280d5f8d815%20png"
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Suppose the incircle touches the sides AB, BC and CA of ∆ABC at D, E and F respectively
Now, R is the centre of incircle
Join RD, RE and RF
Now, RD = RE = RF = radii of incircle
We know that tangent to the circle is always ⊥ to its radius at the point of contact, soRD⊥AB; RE⊥BC; RF⊥CA
Now, we have, AB = 12 cm; BC = 20 cmNow, AC = BC2 - AB2   Pythagoras theorem⇒AC = 202-122 = 400-144 = 256 = 16 cmNow, area of ∆ABC = 12×base×height = 12×AB×AC = 12×12×16 = 96 cm2Now, join RC, RA and RB
ar∆ARC + ar∆ARB + ar∆BRC = area of ∆ABC⇒12×AC×RF + 12×AB×RD +12×BC×RE = 96⇒12×16×RF + 12×12×RF +12×20×RF = 96⇒8RF + 6RF + 10RF = 96⇒24RF = 96⇒RF = 4 cmSo, radius of incircle, r = 4 cmNow, area of incircle = πr2 = 227×42 = 3527cm2

In right triangle ABC angle A =90°. AB = 12cm and BC =20cm. R is the way incentre of triangle ABC. Find the area of the circle drawn with R as centre touching the sides of the triangle.