MHT CET · Maths · Properties of Triangles
With usual notations, if in \(\triangle \mathrm{ABC}, 8\) is semi perimeter and \((\mathrm{s}-a)(\mathrm{s}-\mathrm{b})=\mathrm{s}(\mathrm{s}-\mathrm{c})\),
then \(\triangle A B C\) is
- A an equilateral triangle
- B an obtuse angle triangle
- C a right angled triangle
- D an acute angle triangle
Answer & Solution
Correct Answer
(C) a right angled triangle
Step-by-step Solution
Detailed explanation
We have
\(\begin{array}{l}
\sin \frac{C}{2}=\sqrt{\frac{(s-a)(s-b)}{a b}} \Rightarrow \sin ^{2} \frac{C}{2}=\frac{(s-a)(s-b)}{a b} \text { and } \\
\cos \frac{C}{2}=\sqrt{\frac{s(s-c)}{a b}} \Rightarrow \cos ^{2} \frac{C}{2}=\frac{s(s-c)}{a b} \\
\text { Given }(s-a)(s-b)=s(s-c) \\
\therefore \quad a b \sin ^{2} \frac{C}{2}=a b \cos ^{2} \frac{C}{2}
\end{array}\)
\(\therefore \tan ^{2} \frac{C}{2}=1 \Rightarrow \tan \frac{C}{2}=1 \Rightarrow \frac{C}{2}=45^{\circ} \Rightarrow C=90^{\circ}\)
\(\therefore \triangle \mathrm{ABC}\) is a right angled triangle
\(\begin{array}{l}
\sin \frac{C}{2}=\sqrt{\frac{(s-a)(s-b)}{a b}} \Rightarrow \sin ^{2} \frac{C}{2}=\frac{(s-a)(s-b)}{a b} \text { and } \\
\cos \frac{C}{2}=\sqrt{\frac{s(s-c)}{a b}} \Rightarrow \cos ^{2} \frac{C}{2}=\frac{s(s-c)}{a b} \\
\text { Given }(s-a)(s-b)=s(s-c) \\
\therefore \quad a b \sin ^{2} \frac{C}{2}=a b \cos ^{2} \frac{C}{2}
\end{array}\)
\(\therefore \tan ^{2} \frac{C}{2}=1 \Rightarrow \tan \frac{C}{2}=1 \Rightarrow \frac{C}{2}=45^{\circ} \Rightarrow C=90^{\circ}\)
\(\therefore \triangle \mathrm{ABC}\) is a right angled triangle
See the Complete Solution
Get step-by-step explanations for this and 2.5 Lakh+ more JEE, NEET & CET questions.
- Unlock all solutions
- Practice the full chapter
- Track accuracy across PYQs
4.8 rated on Google Play · 14,000+ reviews
More questions from Maths
- Negation of a statement 'If \(\forall x, x\) is a complex number then \(x^2<0\) ' isMHT CET 2022 Easy
- The value of the integral \(\int_{\frac{-\pi}{2}}^{\frac{\pi}{2}}\left(x^2+\log \frac{\pi-x}{\pi+x}\right) \cos x d x\) is equal toMHT CET 2024 Medium
- The value of \(\lim _{x \rightarrow \infty}\left(\frac{x^{2}-2 x+1}{x^{2}-4 x+2}\right)^{x}\) isMHT CET 2007 Easy
- If \(a^2+b^2+c^2=r^2\), then the value of \(\tan ^{-1}\left(\frac{a b}{c r}\right)+\tan ^{-1}\left(\frac{b c}{a r}\right)+\tan ^{-1}\left(\frac{c a}{b r}\right)=\)MHT CET 2025 Medium
- In a \(\triangle \mathrm{PQR}, \mathrm{m} \angle \mathrm{R}=\frac{\pi}{2}\). If \(\tan \left(\frac{\mathrm{P}}{2}\right)\) and \(\tan \left(\frac{\mathrm{Q}}{2}\right)\) are the roots of the equation \(a x^2+b x+c=0(a \neq 0)\), thenMHT CET 2024 Medium
- It is required to seat 5 men and 4 women in a row so that the men occupy odd places. Then the number of arrangements that are possible isMHT CET 2022 Easy
More PYQs from MHT CET
- The temperature of an ideal gas is increased from 100 K to 400 K. If ' x ' is the R.M.S. velocity of its molecules at 100 K, it becomesMHT CET 2025 Easy
- In a triangle \(\mathrm{ABC}, \mathrm{m} \angle \mathrm{A}, \mathrm{m} \angle \mathrm{B}, \mathrm{m} \angle \mathrm{C}\) are in A.P. and lengths of two larger sides are 10 units, 9 units respectively, then the length (in units) of the third side isMHT CET 2023 Medium
- If are cartesian co-ordinates of the point, then its polar co-ordinates are…..MHT CET 2019 Easy
- In a vessel, the ideal gas is at a pressure \(\mathrm{P}\). If the mass of all the molecules is halved and their speed is doubled, then resultant pressure of the gas will beMHT CET 2023 Medium
- The order of reaction for which the units of rate constant are \(\mathrm{mol} \mathrm{dm}{ }^{-3} \mathrm{~s}^{-1}\) isMHT CET 2021 Medium
- Consider the Doppler effect in two cases. In the first case, an observer moves towards a stationary source of sound with a speed of \(50 \mathrm{~m} / \mathrm{s}\). In the second case, the observer is at rest and the source moves towards the observer with the same speed of \(50 \mathrm{~m} / \mathrm{s}\). Then the frequency heard by the observer will be [velocity of sound in air \(=330 \mathrm{~m} / \mathrm{s}\).]MHT CET 2023 Easy