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.
After having gone through the stuff given above, we hope that the students would have understood "Converse of the pythagorean theorem".
Apart from the stuff given above, if you want to know more about "Converse of the pythagorean theorem proof", please click here
Apart from the stuff given in this section, if you need any other stuff in math, please use our google custom search here.
APTITUDE TESTS ONLINE
ACT MATH ONLINE TEST
TRANSFORMATIONS OF FUNCTIONS
ORDER OF OPERATIONS
MATH FOR KIDS
HCF and LCM word problems