Question

# Write a coordinate proof.

Any right isosceles triangle can be subdivided into a pair of congruent right isosceles triangles.(Hint: Draw the segment from the right angle to the midpoint of the hypotenuse.)

Hint:

### Prove using congruence criterion.

## The correct answer is: any right isosceles triangle can be subdivided into a pair of congruent right isosceles triangles is proved.

### Complete step by step solution:

Let ABD be the right triangle right angled at A. Here BD is the hypotenuse and

C is the midpoint of the hypotenuse so that BC= DC.

Here, AB = AD (since it is isosceles triangle)

∠DAB = 90°

∠ADB = ∠ABD = 45^{0} (since the base angles of an isosceles triangles are equal)

By the converse of the midpoint theorem, P is the midpoint of AD.

∠DPC = ∠DAB = 90°(since PC ∥ AB)

Consider ⃤ DPC and ⃤ APC,

DP = AP (since P is the midpoint)

∠DPC = ∠APC = 90°

CP = CP (common side)

∴ ⃤ DPC and ⃤ APC are congruent by SAS congruence criterion.

_{ ⇒ }_{DC = AC} (corresponding parts of congruent triangles)

So ⃤ ACD is an isosceles triangle.

∠CDA = ∠DAC = 45^{0} (since the base angles of an isosceles triangles are equal)

⇒∠ACD = 90°

Likewise, AC = CB

So ⃤ ACB is an isosceles triangle.

∠CBA = ∠CAB = 45^{0} (since the base angles of an isosceles triangles are equal)

⇒∠ACDB= 90°

Consider 2 triangles ⃤ ACD and ⃤ ACB

∠ACD =∠ACB= 90°

CA = CA (common side)

CD=CB (C is the midpoint)

Hence ⃤ ACD and ⃤ ACB are congruent by SAS congruence criterion.

Thus any right isosceles triangle can be subdivided into a pair of congruent right

isosceles triangles is proved.

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