MHB Find the radius of the sector adjoining a triangle

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The discussion focuses on finding the radius of a sector adjoining a triangle, where the area of the sector equals the area of the triangle. The formula used for the area of the sector is derived, leading to the equation involving the radius, r. After solving, it is determined that the radius r is 9 cm. Participants confirm the correctness of this solution through verification of the area calculations. The final consensus is that r equals 9 cm, with suggestions to use the π symbol for accuracy in calculations.
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We know that the area of the sector should be $\frac{40}{360}$*$\frac{22}{7}$*$r$*r

Any ideas on how to begin?

Many Thanks:)
 
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We know the area of the triangle and the area of the sector are equal, so using the respective formula for those areas we may state:

$$\frac{1}{2}(2\pi)(r)=\frac{1}{2}\left(40^{\circ}\cdot\frac{\pi}{180^{\circ}}\right)r^2$$

What do you get when solving for $r$?
 
MarkFL said:
We know the area of the triangle and the area of the sector are equal, so using the respective formula for those areas we may state:

$$\frac{1}{2}(2\pi)(r)=\frac{1}{2}\left(40^{\circ}\cdot\frac{\pi}{180^{\circ}}\right)r^2$$

What do you get when solving for $r$?

$\displaystyle \frac{1}{2}(2\pi)(r)=\frac{1}{2}\left(40^{\circ}\cdot\frac{\pi}{180^{\circ}}\right)r^2$

$\displaystyle (\pi)(r)=\left(\frac{\pi}{9^{\circ}}\right)r^2$

$9 \displaystyle (\pi)(r)=\pi r^2$

Now Let's use factorization to find r ,

$9 \displaystyle (\pi)(r)=\pi r^2$

$9 \displaystyle (\pi r)=(\pi r) * r $

$9 cm =r$

Now to check whether It is correct,

MarkFL said:
We know the area of the triangle and the area of the sector are equal

$ \displaystyle \frac{1}{2} * 2 * \frac{22}{7} * 9 = \frac{22}{7} * \frac{40}{360}* 9^2$

$ \displaystyle \frac{22}{7} * 9 = \frac{22}{7} * \frac{1}{9}* 9^2$

$ \displaystyle \frac{22}{7} * 9 = \frac{22}{7} * \frac{1}{9}* 9 * 9$

$ \displaystyle \frac{22}{7} * 9 = \frac{22}{7} * 9 $

Correct I guess ? :)

Many Thanks :)
 
Last edited:
Yes, I also got:

$$r=9\text{ cm}$$

In your second line, the degrees would have "cancelled" and so you would just have:

$$(\pi)(r)=\left(\frac{\pi}{9}\right)r^2$$

When checking the answer, I would simply use the $\pi$ symbol rather than a rational approximation for $\pi$. :)
 
:) Thanks For the advice.
 
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