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# NMTC Inter Level Problem 8

Two sides of a triangle are $$8$$ cm. and $$18$$ cm. and the bisector of the angle formed by them is of length $$\frac{60}{13}$$ cm. the length of the third side is

Options:

(A) $$22$$

(B) $$23$$

(C) $$24$$

(D) $$25$$

Note by Nanayaranaraknas Vahdam
2 years, 7 months ago

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The formula for the bisector of any triangle is $$l=\frac{2ab \cos\frac{\theta}{2}}{a+b}$$, where $$\theta$$ is the angle between the sides $$a,b$$. Now that you know the cosine of the angle, just use the cosine theorem (also note that $$\cos \theta = 2\cos^2\frac{\theta}{2}-1$$).

Using this method we can see that $$\cos \frac{\theta}{2}=\frac{5}{12}$$, thus $$\cos\theta = 2\cdot \frac{25}{144}-1=\frac{50}{144}-1=-\frac{47}{72}$$, hence by using the cosine theorem we can see that

$$\color{RoyalBlue}{\text{answer}}=\sqrt{64+324+288\cdot \frac{47}{72}}=\sqrt{388+4\cdot 47}=\boxed{(\text{C})\text{ }24}$$. · 2 years, 7 months ago

We have the formula of length of angle bisector as derived here

$$d^2=\dfrac{bc}{(b+c)^2} \Bigl( (b+c)^2-a^2\Bigr)$$

$$\dfrac{60\times 60}{13\times 13} = \dfrac{8\times 18}{26\times 26} (26^2-a^2)$$

$$26^2-a^2=\dfrac{60\times 60\times 26\times 26}{13\times 13\times 18\times 8} = 100$$

This gives $$a^2=576 \implies a=\boxed{24}$$ · 2 years, 7 months ago

What is this formula known as? · 2 years, 7 months ago