Question

Show that if the diagonals of a square are equal and bisect each other at right angle.
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Answer 1

divide the square into 4 triangles, prove them congruent by RHS test, using cpctc, u can prove the equal and at 90°

Answer 2

✨ANSWER✨

Let ABCD be a square such that its diagonals AC and BD intersect at O. We have to prove that AC=BD and AC and BD bisect each other at right angle.

In △BAD and △CDA,

BA=CD

∠BAD=∠CDA

AD=DA

So, by SAS congruence criterion,

△BAD≅△CDA

⟹BD=CA

Since every square is a parallelogram and diagonals of a parallelogram bisect each other.

Therefore,

OA=OC and OB=OD

Now, consider triangles AOB and COB. In these triangles,

AB=CB

OB=OB

OA=OC

So, by SSS congruence criterion,

△AOB≅△COB

∠AOB=∠COB

But, ∠AOB+∠COB=180°

∠AOB=∠COB=90°

Hence, diagonals of square ABCD are equal and bisect each other at right angles.

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