AD is a median of triangleABC if BL and CM are drawn perpendicular on AD and AD produced prove that BL=CM

Dyu

To prove that BL = CM, we need to show that triangles BLC and CMA are congruent. Since AD is a median, it intersects the opposite side BC at its midpoint, denoted as E.

To start the proof, we can establish the right angles at L and M:

1. Since BL is perpendicular to AD, angle BLD = 90 degrees.
2. Similarly, since CM is perpendicular to AD, angle DMC = 90 degrees.

Next, we can use the midpoint property to show that segments BL and CM have equal lengths:

3. Consider triangle BLC. Since D is the midpoint of BC, we have DE = EC (as E is the midpoint of BC as well).
4. Since triangle BLC has right angle BLD, it must be a right-angled triangle.
5. By the Pythagorean theorem, in a right-angled triangle, the squares of the two shorter sides are equal to the square of the hypotenuse: BL^2 = BD^2 + LD^2. Let's call this equation (1).

6. Now consider triangle CMA. With D as the midpoint of BC, we have DE = EC again.
7. Triangle CMA also has a right angle at DMC.
8. By the Pythagorean theorem in triangle CMA, we have CM^2 = CD^2 + MD^2. Let's call this equation (2).

To complete the proof, we need to show that equation (1) is equal to equation (2):

9. Since angle BLD = angle DMC, the two triangles BLD and DMC share an angle.
10. By the angle-angle criterion for triangle congruence, we can conclude that the triangles BLD and DMC are congruent.
11. As a result, BD = CD and LD = MD in equations (1) and (2), respectively.
12. Therefore, equation (1) reduces to BL^2 = CM^2, or BL = CM.

Hence, we have proven that BL = CM using the properties of a median and the congruence of triangles BLC and CMA.