Question

Let PQRS be a quadrilateral. Two circles $O_1$ and $O_2$ are inscribed in $\Delta PQR$ and $\Delta PSR$, respectively. Circle $O_1$ touches PR at M and circle $O_2$ touches PR at N. Find the length of MN.
I. A circle is inscribed in the quadrilateral PQRS.
II. The radii of the circles $O_1$ and $O_2$ are 5 and 6 units respectively.

• A. Statement I alone is sufficient to answer the questions
• 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

Answer 1

The correct option is A Statement I alone is sufficient to answer the questions

In the first quadrilateral

Let PD = a, QD = b, RM = c, PA = d and SA = e
Now, PA = PN + MN, QD = QC
RC = RN + RM, RM = RB and SD = SB
PS = d + e, SR = e + c, QR = b + c + MN, PQ = a + b
In the second quadrilateral,
PQ + SR = PS + QR
$x_1+x_4+x_2+x_3=x_1+x_2+x_3+x_4$
a + b + e + c = d + e + b + c + MN
a = a + MN + MN ($\therefore$ a + d + MN)
2MN = 0 $\Rightarrow$ MN = 0
So, the measure of MN can be answered from Statement I alone. Thus, Statement I alone is sufficient. Statement II alone is not sufficient as we can have more than one value of MN possible.