in triangle ABC, angle C=90 degrees, and M is the midpoint of AB. by using INDIRECT PROOF, prove that Cm=AM=BM

Answer 1

If angle c of triangle ABC is 90 degrees that means it's a right angle triangle. The Pythagoras theorem applies. Alternatively, If you draw an imaginary line from M to C it will dissect the angle into two equal parts of 45 degrees each and would be perpendicular to AB. Now the sum of all the angles of a triangle is 180 degrees. So with the perpendicular you have the 90 degree angle plus the 45 angle which leaves the other angle which has to be 45. Hence all triangles are right angle isoceles triangles and sides corresponding to 45 angles that is AM=CM=BM. Also AC is equal to BC.

Answer 2

If CM not = AM or CM then it must be larger or smaller than each by the law of trichotomy.
Assume CM > AM
Then angle A > angle ACM , and angle B > angle BCM
Thus angle A +angle B > angle ACM+ angle BCM
But angle ACM+ angle BCM =angle C =90 degrees
Then the sum of the angles of triangle ABC > 180 degrees.
But this is impossible so AM not > BM
The same argument can be made if we assume CM < AM =BM

Hence CM = AM = BM

Answer 3

in indirect proofs, you have to make an assumption. If your assumption turns out to be false then the opposite must be true

Answer 4

The sol. to you question is given in S.Chand's Textbook of Maehematics, pg no. 321