Question:
Given the vertices of a triangle are \( A(1, -1, -3) \), \( B(2, 1, -2) \), and \( C(-5, 2, -6) \). Compute the length of the bisector of the interior angle at vertex A.
Given the vertices of a triangle are \( A(1, -1, -3) \), \( B(2, 1, -2) \), and \( C(-5, 2, -6) \). Compute the length of the bisector of the interior angle at vertex A.
Show Hint
For finding the angle bisector length in 3D, use the general formula for bisectors in triangles with known side lengths.
Updated On: Apr 1, 2025
- 3
- \( \frac{\sqrt{10}}{4} \)
- \( 3 \sqrt{10} \)
- \( 4 \)
Hide Solution
Verified By Collegedunia
The Correct Option is C
Solution and Explanation
The formula for the length of the angle bisector is:
\[
l = \sqrt{bc \left( 1 - \frac{a^2}{(b+c)^2} \right)}
\]
where \( a \), \( b \), and \( c \) are the sides of the triangle.
After computing the side lengths and substituting into the formula, we find that the length of the bisector is \( 3 \sqrt{10} \).
Was this answer helpful?
0
0
Top Questions on 3D Geometry
- Let P be the foot of the perpendicular from the point \( (1, 2, 2) \) on the line \[ \frac{x-1}{1} = \frac{y + 1}{-1} = \frac{z - 2}{2} \] Let the line \( \mathbf{r} = (-\hat{i} + \hat{j} - 2\hat{k}) + \lambda (\hat{i} - \hat{j} + \hat{k})\), \( \lambda \in \mathbb{R} \), intersect the line \(L\) at \(Q\). Then \( 2(PQ)^2 \) is equal to:
- JEE Main - 2025
- Mathematics
- 3D Geometry
- The square of the distance of the point \(\left( \frac{15}{7}, \frac{32}{7}, 7 \right)\) from the line \(\frac{x+1}{3} = \frac{y+3}{5} = \frac{z+5}{7}\) in the direction of the vector \(\mathbf{i} + 4\mathbf{j} + 7\mathbf{k}\) is:
- JEE Main - 2025
- Mathematics
- 3D Geometry
- Let in a \( \triangle ABC \), the length of the side AC is 6, the vertex B is \( (1, 2, 3) \) and the vertices A, C lie on the line \[ \frac{x - 6}{3} = \frac{y - 7}{2} = \frac{z - 7}{-2}. \] Then the area (in sq. units) of \( \triangle ABC \) is:
- JEE Main - 2025
- Mathematics
- 3D Geometry
- Let \( L_1 : \frac{x - 1}{3} = \frac{y - 1}{-1} = \frac{z + 1}{0} \) and \( L_2 : \frac{x - 2}{2} = \frac{y}{0} = \frac{z + 4}{\alpha} \), where \( \alpha \in \mathbb{R} \), be two lines which intersect at the point \( B \). If \( P \) is the foot of the perpendicular from the point \( A(1, 1, -1) \) on \( L_2 \), then the value of \( 26 \alpha (PB)^2 \) is:
- JEE Main - 2025
- Mathematics
- 3D Geometry
- Let a straight line \( L \) pass through the point \(P(2, -1, 3)\) and be perpendicular to the lines \[ \frac{x - 1}{2} = \frac{y + 1}{1} = \frac{z - 3}{-2} \quad \text{and} \quad \frac{x - 3}{1} = \frac{y - 2}{3} = \frac{z + 2}{4}. \] If the line \(L\) intersects the yz-plane at the point Q, then the distance between the points P and Q is:
- JEE Main - 2025
- Mathematics
- 3D Geometry
View More Questions
Questions Asked in IPU CET exam
Which of the following is an octal number equal to decimal number \((896)_{10}\)?
- IPU CET - 2023
- Binary Operations
The additional 8% human genome sequenced account for ........ million new letters added to the existing sequenced DNA.
- IPU CET - 2023
- Genomics, proteomics, metabolomics
- Which of the following statements are true or false?
I. Endemic when it spreads across the globe is called a pandemic.
II. Epidemic is a disease that is permanently present in a region or population
Which of the following options is correct?- IPU CET - 2023
- Epidemiology
- Which one among the following is not a characteristic of 'big data'?
- IPU CET - 2023
- Big Data
- Blockchain Technology has the following hall mark features/s:
- IPU CET - 2023
- Blockchain Technology
View More Questions