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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


(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

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

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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 · 2 years, 7 months ago

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@Aditya Raut What is this formula known as? Saurabh Mallik · 2 years, 7 months ago

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@Saurabh Mallik idk, Length of angle bisector,maybe ! Name is not important, formula is important ! Aditya Raut · 2 years, 6 months ago

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