Converse of the pythagorean theorem proof :
Already we know the Pythagorean Theorem for right triangles. In this section, we are going to see the converse of the Pythagorean theorem.
The Pythagorean Theorem states that if a triangle is a right triangle, then, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse.
That is, if a and b are legs and c is the hypotenuse, then
a² + b² = c²
The converse of the Pythagorean Theorem states that if a² + b² = c², then the triangle is a right triangle.
Converse of the Pythagorean Theorem :
In a triangle, if the square of one side is equal to the sum of the squares of the other two sides, then the angle opposite to the first side is a right angle.
A triangle ABC such that AB² + BC² = AC²
To Prove :
ΔABC is right-angled at B.
Construct a right-angled triangle PQR, right-angled at Q such that PQ = AB and QR = BC.
Step 1 :
In ΔPQR, ∠Q = 90°.
Using Pythagorean theorem in ΔPQR, we have
PQ² + QR² = PR² ----- (1)
Step 2 :
In ΔABC (given), we have
AB² + BC² = AC² ----- (2)
Step 3 :
By construction, PQ = AB and QR = BC.
So, from (1) and (2), we have
PR² = AC²
Get rid of the square from both sides.
PR = AC
Step 4 :
Therefore, by SSS congruence criterion, we get
ΔABC ≅ ΔPQR
∠B = ∠Q
Step 5 :
But, we have ∠Q = 90° by construction.
Therefore ∠B = 90°.
Hence, ΔABC is a right triangle, right angled at B.
Thus, the theorem is proved.
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