bhandariavi wrote:

In the figure (attached) here, lines l and m are parallel. If the arc AB = 3π, and the ratio of PA to PQ is 2/3, what is the length of m?

Choices

A

15 * Sqroot2

B

8 * Sqroot2

C

15

D

9pi

E

9 * Sqroot2

Let's name the other vertex of line m as R.

\(\angle{QPR}=\angle{APB}\)

\(\angle{PQR}=\angle{PAB}\) \({l || m}\)

\(\angle{PRQ}=\angle{PBA}\) \({l || m}\)

According to AAA similarity criterion, \(\triangle{QPX}\approx\triangle{APB}\)

Means QPX and APB are similar triangle.

The ratio will be same for the sides of two similar triangles;

\(\frac{l}{m}=\frac{PA}{PQ}=\frac{2}{3}\)

Now, let's get to finding the length of l.

The angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle.

Meaning thereby; arc AB will subtend an angle of 90 degree with the center.

Length of an arc;

\(\frac{\theta}{360}*2*\pi*r=3*\pi\)

\(\frac{90}{360}*2*\pi*r=3*\pi\)

\(r=6\)

Another interesting thing to note is that; if arc AB subtended an angle of 90 degree with the center, line segment AB becomes the hypotenuse of a right angled triangle with sides as radius.

Thus;

\((AB)^2=6^2+6^2\)

\(l=\sqrt{72}\)

\(l=6\sqrt{2}\)

Substitute in the first equation;

\(\frac{l}{m}=\frac{2}{3}\)

\(\frac{6\sqrt{2}}{m}=\frac{2}{3}\)

\(m=9\sqrt{2}\)

Ans: "E"

_________________

~fluke

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