Question:
In \( \triangle ABC \), \( \frac{b - c}{r_1} + \frac{c - a}{r_2} + \frac{a - b}{r_3} \) is equal to
In \( \triangle ABC \), \( \frac{b - c}{r_1} + \frac{c - a}{r_2} + \frac{a - b}{r_3} \) is equal to
Show Hint
The sum \( \frac{b - c}{r_1} + \frac{c - a}{r_2} + \frac{a - b}{r_3} = 0 \) is a standard identity in triangle geometry that holds for all triangles.
Updated On: Apr 22, 2026
- 0
- \( abc \)
- \( a + b + c \)
- \( ab + bc + ca \)
Show Solution
Verified By Collegedunia
The Correct Option is A
Solution and Explanation
Step 1: Understanding the equation.
We are given the equation \( \frac{b - c}{r_1} + \frac{c - a}{r_2} + \frac{a - b}{r_3} \), where \( r_1, r_2, r_3 \) are the radii of the excircles opposite to the vertices \( A, B, C \) respectively.
Step 2: Use of known identity.
There is a well-known identity in triangle geometry which relates the terms involving the excircles and the sides of the triangle: \[ \frac{b - c}{r_1} + \frac{c - a}{r_2} + \frac{a - b}{r_3} = 0 \] This identity holds for any triangle.
Step 3: Simplification.
From the identity, it directly follows that: \[ \frac{b - c}{r_1} + \frac{c - a}{r_2} + \frac{a - b}{r_3} = 0 \]
Step 4: Conclusion.
Thus, the value of the given expression is 0, corresponding to option (A).
We are given the equation \( \frac{b - c}{r_1} + \frac{c - a}{r_2} + \frac{a - b}{r_3} \), where \( r_1, r_2, r_3 \) are the radii of the excircles opposite to the vertices \( A, B, C \) respectively.
Step 2: Use of known identity.
There is a well-known identity in triangle geometry which relates the terms involving the excircles and the sides of the triangle: \[ \frac{b - c}{r_1} + \frac{c - a}{r_2} + \frac{a - b}{r_3} = 0 \] This identity holds for any triangle.
Step 3: Simplification.
From the identity, it directly follows that: \[ \frac{b - c}{r_1} + \frac{c - a}{r_2} + \frac{a - b}{r_3} = 0 \]
Step 4: Conclusion.
Thus, the value of the given expression is 0, corresponding to option (A).
Was this answer helpful?
0
0
Top MET Mathematics Questions
- $\int\limits_{0}^{8}| x -5| d x$ is equal to
- The equation $3^{3x + 4} = 9^{2x - 2},\ x>0$ has the solution
- A parallelogram is constructed on the vectors $\mathbf{a} = 3\alpha - \beta$, $\mathbf{b} = \alpha + 3\beta$. If $|\alpha| = |\beta| = 2$ and the angle between $\alpha$ and $\beta$ is $\dfrac{\pi}{3}$, then the length of a diagonal of the parallelogram is
- Every group of order 7 is
- Let $U_{n} = 2 + 2^{3} + 2^{5} + \cdots + 2^{2n+1}$ and $V_{n} = 1 + 4 + 4^{2} + \cdots + 4^{n-1}$. Then $\displaystyle\lim_{n \to \infty} \dfrac{U_n}{V_n}$ is equal to
View More Questions
Top MET Trigonometry Questions
- If $\tan^{-1}x + \tan^{-1}y + \tan^{-1}z = \pi$, then $x + y + z$ is
- If $3\sin^{-1}\!\left(\dfrac{2x}{1+x^{2}}\right) - 4\cos^{-1}\!\left(\dfrac{1-x^{2}}{1+x^{2}}\right) + 2\tan^{-1}\!\left(\dfrac{2x}{1-x^{2}}\right) = \dfrac{\pi}{3}$, then $x$ equals
- If $\sin(\pi \cos \theta) = \cos(\pi \sin \theta)$, then which of the following is correct?
- $\sin^{-1}\dfrac{1}{\sqrt{5}} + \cos^{-1}\dfrac{3}{\sqrt{10}}$ is equal to
- If $4\sin^{-1}x + \cos^{-1}x = \pi$, then $x$ is equal to
View More Questions
Top MET Questions
- A coil of 100 turns and area $2 \times 10^{-2}m^2 $ is pivoted about a vertical diameter in a uniform magnetic field and carries a current of 5A. When the coil is held with its plane in north-south direction, it experiences a couple of 0.33 Nm. When the plane is east-west, the corresponding couple is 0.4 Nm, the value of magnetic induction is [Neglect earth's magnetic field]
- $\int\limits_{0}^{8}| x -5| d x$ is equal to
- Radiation, with wavelength 6561 $?$ falls on a metal surface to produce photoelectrons. The electrons are made to enter a uniform magnetic field of $3 \times 10^{-4} T$. If the radius of the largest circular path followed by the electrons is 10 mm, the work function of the metal is close to :
- In intrinsic semiconductor at room temperature number of electrons and holes are
- Which of the following product is formed in the reaction $ CH_3MgBr {->[(i)CO_2][(ii)H_2O]} ? $
View More Questions