Question

Point M is the midpoint of AB . Through point M a line segment XY is drawn, so that both XA and YB are perpendicular to AB . Prove that AY ≅ BX .

Answer 1

Answer:

See the proof below

Step-by-step explanation:

Let the line AB be a straight line on the parallelogram.

A dissection of the line (using the perpendicular line X) gives:

AY ≅ BX

Another way will be using the angles.

The angles are equal - vertically opposite angles

Hence the line  AY ≅ BX (Proved)

Answer 2

Answer:

AY = BX proved.

Step-by-step explanation:

See the diagram attached to the answer.

Given AM = MB and ∠BAX = ∠ABY = 90°

Now, between ΔAXM and ΔBYM,

(i) AM = BM {Given}

(ii) ∠XAM = ∠YBM {Given}

(iii) ∠AMX = ∠BMY {Opposite angles}

So, we can say that ΔAXM ≅ ΔBYM

Hence, AX = BY.......... (1)

Now, between ΔABX and ΔABY,

(i) AB is common side.

(ii) AX = BY {Already proved} and

(iii) ∠BAX = ∠ABY {Given}

Therefore, ΔABX ≅ ΔABY.

Hence, BX = AY {Corresponding sides} (Proved)