Chapter 8: Q. 62 (page 531)

Circumscribing a TriangleShow that $\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c} = \frac{1}{2 r}$
where $r$ is 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 that $\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c} = \frac{1}{2 r}$

Step by step solution

Step 1. Given Information

We have

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

Step 2. Identifying angles

First, notice that angles $∠ P Q R$ and $∠ P P' R$ are both inscribed angles.

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

This means that they are equal. $∠ P Q R = ∠ P P' R$

Since $∠ P Q R = B, t h e n a l s o ∠ P P' R = B$

Step 3. Proving the result

Since $\sin (∠ P P' R) = \frac{b}{2 r} \frac{\sin (∠ P P' R)}{b} = \frac{1}{2 r} \frac{\sin B}{b} = \frac{1}{2 r} [S i n c e ∠ P P' R = B]$

Now, role="math" localid="1646590875136" $\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c} a n d \frac{\sin B}{b} = \frac{1}{2 r}$

then $\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c} = \frac{1}{2 r}$

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Circumscribing a TriangleShow that $\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c} = \frac{1}{2 r}$
where $r$ is the radius of the circle circumscribing the triangle PQR whose sides are a, b, and c, as shown in the figure.

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Identifying angles

Step 3. Proving the result

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Circumscribing a TriangleShow that sinAa=sinBb=sinCc=12rwhere ris the radius of the circle circumscribing the triangle PQR whose sides are a, b, and c, as shown in the figure.

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Identifying angles

Step 3. Proving the result

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Statistics

Theoretical and Mathematical Physics

Pure Maths

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Study anywhere. Anytime. Across all devices.

Circumscribing a TriangleShow that $\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c} = \frac{1}{2 r}$
where $r$ is the radius of the circle circumscribing the triangle PQR whose sides are a, b, and c, as shown in the figure.