BM and CN are perpendicular to a line passing through the vertex A of a triangle ABC. IF L is the mid point of BC. PROVE THAT,LM=LN
To prove that LM = LN, we can use the concept of the midpoint theorem and the fact that BM and CN are perpendicular to the line passing through the vertex A.
First, recall the midpoint theorem:
In a triangle, the line drawn from a vertex to the midpoint of the opposite side is half as long as the line drawn from that vertex to the corresponding side.
Using this theorem, we can conclude that AM = AN, where M and N are the midpoints of AB and AC, respectively.
Now, let's consider triangle ABC with perpendicular lines BM and CN.
Since L is the midpoint of BC, we have LM = LC and LN = LB, as L is also the midpoint of BN and CM.
Now, let's consider the triangles LBM and LCN.
Since BM and CN are perpendicular to the same line passing through vertex A, LM is parallel to CN, and LN is parallel to BM.
By using the Midpoint Theorem, we know that AM = AN and AL = AL (common side).
Therefore, we can conclude that triangle LBM is similar to triangle LCN (by AA similarity, as two angles are 90 degrees and the third pair of angles LBM and LCN are congruent, since LM || CN and LN || BM).
Since the corresponding sides of similar triangles are proportional, we have:
LM/LN = LB/LC
But, since LM = LC and LN = LB (as shown earlier), the equation simplifies to:
1/1 = 1/1
Thus, we have proved that LM = LN.
To prove that LM = LN, we need to show that L is the midpoint of MN.
Given that BM and CN are perpendicular to a line passing through the vertex A, we can conclude that the points B, M, C, and N lie on a straight line.
Let's consider the triangle ABC. Since L is the midpoint of BC, we can express the lengths of BM, CM, and LN in terms of LM.
By the midpoint theorem, we know that BM = 2LM. (1)
Similarly, CM = 2LM. (2)
Since BM, CN, and LN are collinear, we can express CN in terms of other lengths.
By combining equations (1) and (2), we get:
BM + CM = 2LM + 2LM
BC = 4LM
From the same line, we can express CN in terms of BM:
CN = BC - BN
CN = 4LM - 2LM
CN = 2LM
Now, we can compare LN and LM:
LN = CN + CM
LN = 2LM + 2LM
LN = 4LM
Since we have shown that LN = 4LM, and we have also proven that BC = 4LM, we can conclude that L is indeed the midpoint of MN. Therefore, LM = LN.