Question

In the given figure, the perimeter of the quadrilateral APOQ is 26 cm and AP $ = 10cm$. O is the center of the circle. Then the radius of the circle is (where AP and AQ are the tangents from A)
(A) 6 cm (B) 3 cm (C) 8 cm (D) 16 cm

Answer 1

Hint: The perimeter of the quadrilateral will be the sum of its all sides. Start with assuming some variable for the radius of the circle and then find the perimeter in terms of that variable. Then solve it using the data given in the question.

Complete step-by-step answer:
According to the information given in the question, perimeter of APOQ is 26 cm.
We know that, perimeter is the sum of all sides of the quadrilateral. Using this we have:
$\therefore $$AP + AQ + OP + OQ = 26 .....(i)$,
Now, OP and OQ are the radius of the circle. Let the radius of the circle be r, then:
$OP = OQ = r$
And the lengths of AP and AQ will be same because they are tangents drawn from the same point to the circle. Length of AP is 10 cm as given in the question. So we have:
$AP = AQ = 10$
Now, putting all the values in equation \[(i)\], we’ll get:
$
   \Rightarrow 10 + 10 + r + r = 26, \\
   \Rightarrow 20 + 2r = 26, \\
   \Rightarrow 2r = 6, \\
   \Rightarrow r = 3 \\
$
Thus the radius of the circle is 3 cm.

Note: Lengths of two tangents drawn from an external point to a circle are always equal. Therefore in the above case, line AO is dividing the quadrilateral APOQ into two congruent triangles APO and AQO.