Question

Did i write the correct answer? ​

Answer 1

`\sf\\\textsf{No your answer is wrong. }\\\textsf{First of all, these triangles are not congruent by S.A.S axiom, but rather they}\\\textsf{are congruent by R.H.S. axiom or A.A.S axiom.}\\\textsf{Second, your answer is incomplete. Because you need to prove that`

`\textsf{Here is the corrected version of your answer:}`

`\sf\\\textsf{Solution: }\\\textsf{Given: }\textsf{C is the centre of circle. }CM\perp AB\\\textsf{To prove: AM = BM}\\\textsf{Proof:}`

`\sf\\\textsf{Consider }\triangle AMC\textsf{ and }\triangle CMB.\\1\\(i)\ \angle AMC=\angle CMB=90^o\ (R)\ \ \ \textsf{[CM}\perp AB,\ given]\\(ii)\ AC=BC\ (H)\ \ \ \textsf{[Radii of same circle are equal.]}\\(iii)\ CM=CM\ (S)\ \ \ \textsf{[Common side.]}\\(iv)\ \triangle AMC \cong \triangle CMB\ \ \ [\textsf{By R.H.S. axiom}]`

`\sf\\2.\ AM=BM\ \ \ \textsf{[Corresponding sides of congruent triangles are equal.]}`

`\sf\\\textsf{The reason }\triangle AMC\textsf{ and }\triangle CMB\textsf{ are not congruent by S.A.S. is because congruency}\\\textsf{by S.A.S. means that the first two corresponding sides should be equal, then the }\\\textsf{angles, and then the other two corresponding sides.}`

`\sf\\\textsf{In this case, if you knew that }\angle ACM=\angle BCM, \textsf{ then you could have told that }\\\textsf{the two triangles are congruent.}`

`\sf\\\textsf{Of course, you may say that }\angle ACM=\angle CBM\textsf{ because }AC=BC\textsf{ makes triangle }\\\textsf{ABC an isosceles triangle and }CM\perp AB.\textsf{ In an isosceles triangle, if a line passing}\\\textsf{through the vertex angle is perpendicular to it's base, then the line bisects the base}\\\textsf{and the vertex angle. }`

`\sf\\\textsf{Rather, you may say that }\triangle AMC\cong \triangle CMB \textsf{ by A.A.S. axiom. Here is the way:}`

`\sf\\1.\ AC=BC\ \ \ [\textsf{Radii of same circle.}]\\2.\\(i)\ \angle CAM=\angle CBM\ (A)\ \ \ [AC=BC,\textsf{ triangle ABC is isosceles.}]\\(ii)\ \angle AMC=\angle BMC\ (A)\ \ \ [CM\perp AB, \textsf{ given.}]\\(iii)\ CM=CM\ (S)\ \ \ \textsf{[Common side.]}\\(iv)\ \triangle AMC\cong \triangle CMB\ \ \ [\textsf{By A.A.S. axiom.}]`

Hope this helps!