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) $23$

(D) $25$

#Geometry #NMTC

Easy Math Editor

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Comments

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

$\color{royalblue}{\text{answer}}=\sqrt{64+324+288\cdot \frac{47}{72}}=\sqrt{388+4\cdot 47}=\boxed{(\text{C})\text{ }24}$ .

mathh mathh - 6 years, 9 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 $a^2=576 \implies a=\boxed{24}$

Aditya Raut - 6 years, 9 months ago

What is this formula known as?

Saurabh Mallik - 6 years, 9 months ago

idk, Length of angle bisector,maybe ! Name is not important, formula is important !

Aditya Raut - 6 years, 9 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$