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Easy

Question

ABC is a right-angled triangle with AB=8cm and BC= 6 cm. A circle with centre O has been inscribed inside the triangle. Calculate the value of r the radius of the inscribed circle.

hintHint:

Pythagoras' theorem states that “In a right-angled triangle,  the square of the hypotenuse side is equal to the sum of squares of the other two sides“. The sides of this triangle have been named Perpendicular, Base and Hypotenuse. Here, the hypotenuse is the longest side, as it is opposite to the angle 90°.
If a is the perpendicular, b is the base, and c is the hypotenuse, then according to the definition, the Pythagoras Theorem formula is given as
c2= a2 + b2
The lengths of two tangents drawn from an external point to a circle are equal.
 

The correct answer is: Hence, the radius of the inscribed circle is 2 cm


                                                         
    Given BC = 6 cm and AB = 8 cm
    Applying Pythagoras theorem in △ABC

    AC2 = BC2 + AB2

    AC2 = 62 + 82

    AC2 = 100

    AC= 10 cm

    Let OP = OR = OQ = r
    We know that two tangents drawn from an external point to a circle are equal in measure.
    So, BP = BQ, CQ = CR and AP = AR
    Also, OP and OQ are radii of the circle. OP ⊥ AB and OQ ⊥ BC and ∠B = 90° (given)
    Hence, we can conclude that BPOQ is a square.
    Thus, BP = BQ = r (sides of a square are equal)
    So, AR = AP = AB – BP

    AR = AP = 8 – r

    CR = CQ = BC – BQ

    CR = CQ = 6 – r

    Now, AR + CR = AC

    8 – r + 6 – r = 10

    14 – 2r = 10

    4 = 2r

    r = 2 cm
    Final Answer:
    Hence, the radius of the inscribed circle is 2 cm

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