are followed by two statements labelled as I and II. Decide if these statements are sufficient to conclusively answer the question. Choose the appropriate answer from the options given below:
A. Statement I alone is sufficient to answer the question.
B. Statement II alone is sufficient to answer the question.
C. Statement I and Statement II together are sufficient, but neither of the two alone is sufficient to answer the question.
D. Either Statement I or Statement II alone is sufficient to answer the question.
E. Neither Statement I nor Statement II is necessary to answer the question.
Let PQRS be a quadrilateral. Two circles O1 and O2 are inscribed in triangles PQR and PSR respectively. Circle O1 touches PR at M and circle O2 touches PR at N. Find the length of MN.
I. A circle is inscribed in the quadrilateral PQRS.
II. The radii of the circles O1 and O2 are 5 and 6 units respectively.
Solution
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Using the property that, tangents from same point to a circle are equal in lengths.
In above quadrilateral, PA = PM + MN, => $$d = a + MN$$ ----------Eqn(I)
RC = RN + NM
=> $$PS = d + e$$
$$SR = e + c$$
$$QR = b + c + MN$$
$$PQ = a + b$$
From statement I :
.PNG)
We can conclude that, $$w + x + y + z = w + x + y + z$$
=> $$(w + z) + (x + y) = (w + x) + (y + z)$$
=> $$PQ + SR = PS + QR$$
Substituting values from above equation, we get :
$$\therefore a + b + e + c = d + e + b + c + MN$$
Using eqn(I),
=> $$a = a + MN + MN$$
=> $$MN = 0$$
Thus, statement I alone is sufficient.
Statement II alone is not sufficient, for we can have more than one value of MN possible.
