In the given figure, ABCABC is a right-angled triangle with AB=7 cmAB=7 cm and AC=9 cmAC=9 cm. A circle with center OO has been inscribed inside the triangle. Calculate the value of rr, the radius of the inscribed circle.
O A B C F D E 7 cm 9 cm


Answer:

2.3 cm2.3 cm

Step by Step Explanation:
  1. Let us join OO to A,B,A,B, and CC and draw ODABODAB, OEBCOEBC and OFCAOFCA.
    O A B C F D r r r E 7 cm 9 cm

    We see that OD,OE,OD,OE, and OFOF are the radius of the circle with center OO.
    OD=OE=OF=r cmOD=OE=OF=r cm

    Also, ABCABC is a right-angled triangle.  Area of ABC=12×AB×AC=12×7 cm×9 cm=31.5 cm2
  2. Let us now find the area of ABC in terms of r. Area of ABC=Area of OAB+Area of OBC+Area of OCA=12×AB×OD+12×BC×OE+12×CA×OF=12×AB×r+12×BC×r+12×CA×r=12(AB+BC+CA)×r=12(Perimeter of ABC)×r
  3. Comparing the area of ABC obtained in step 1 and step 2, we have  Area of ABC=31.5 cm2=12×(AB+BC+CA)×r31.5 cm2=12×(7+BC+9)×r(i)
  4. Applying Pythagoras theorem in ABC, we have BC2=AB2+AC2BC=(7)2+(9)2=11.4 cm
  5. Now, substituting the value of BC in eq (i), we have 31.5=12×(7+11.4+9)×r31.5×2=27.4×rr=31.5×227.4=2.3 cm
  6. Hence, the radius of the inscribed circle is 2.3 cm.

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