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Chapter 8: Q. 62 (page 531)

Circumscribing a TriangleShow that sinAa=sinBb=sinCc=12r

where ris the radius of the circle circumscribing the triangle PQR whose sides are a, b, and c, as shown in the figure.

Short Answer

Expert verified

It is proved thatsinAa=sinBb=sinCc=12r

Step by step solution

01

Step 1. Given Information

We have

Let P' be a point on the circumference such thatPP'=2r

02

Step 2. Identifying angles

First, notice that angles 鈭�PQRand 鈭�PP'Rare both inscribed angles.

Also, they both originate from the same two points P and R.

This means that they are equal.鈭�PQR=鈭�PP'R

Since鈭�PQR=B,thenalso鈭�PP'R=B

03

Step 3. Proving the result

Since sin鈭�PP'R=b2rsin鈭�PP'Rb=12rsinBb=12rSince鈭�PP'R=B

Now, role="math" localid="1646590875136" sinAa=sinBb=sinCcandsinBb=12r

thensinAa=sinBb=sinCc=12r

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