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 = 450 (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 = 450 (since the base angles of an isosceles triangles are equal)
⇒∠ACD = 90°
Likewise, AC = CB
So ⃤ ACB is an isosceles triangle.
∠CBA = ∠CAB = 450 (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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