In a triangle ABC right angled at A.the length of altitude from A and length of internal bisector of A are 3 and 4 respectively.find the length of median through A?

Question

Answer

5

Answer

Given information:
- Triangle ABC is right-angled at A.
- Length of altitude from A = 3 units.
- Length of internal bisector of A = 4 units.

To find:
Length of median through A.

Solution:

Step 1: Understanding the problem
- In a triangle, the median is a line segment that connects a vertex to the midpoint of the opposite side.
- We are given the lengths of the altitude from A and the internal bisector of A.
- We need to find the length of the median through A.

Step 2: Drawing the triangle
- Draw a triangle ABC with right angle at A.
- Label the length of the altitude from A as 3 units.
- Label the length of the internal bisector of A as 4 units.

Step 3: Solving the problem
- Since triangle ABC is right-angled at A, the altitude from A divides the triangle into two smaller triangles, let's call them ABD and ACD.
- Let's assume the length of the median through A is x units.
- In triangle ABD, the median divides the side BD into two equal parts, each with length x/2.
- In triangle ACD, the median divides the side CD into two equal parts, each with length x/2.
- The altitude from A divides the side BC into two parts, one with length 3 and the other with length (x-3).
- Using the property of the internal bisector, we can write the equation: (x-3)/4 = 3/4.
- Solving this equation, we get x-3 = 9, which gives x = 12.

Step 4: Answer
- The length of the median through A is 12 units.

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