If the square of the size of the longest side of a triangle is equal to the amount of the squares of the various other 2 sides, then the triangle is a appropriate triangle.

That is, inΔABC, if c2=a2+b2 then∠C is a right triangle,ΔPQR being the appropriate angle.

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We have the right to prove this by contradiction.

Let us assume that c2=a2+b2 in ΔABC and the triangle is not a best triangle.

Now think about one more triangleΔPQR. We constructΔPQR so thatPR=a, QR=b and also ∠R is a appropriate angle.

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By the Pythagorean Theorem, (PQ)2=a2+b2.

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But we recognize that a2+b2=c2 and a2+b2=c2 and c=AB.

So, (PQ)2=a2+b2=(AB)2.

That is, (PQ)2=(AB)2.

Due to the fact that PQ and AB are lengths of sides, we have the right to take positive square roots.

PQ=AB

That is, all the 3 sides of ΔPQR are congruent to the three sides of ΔABC. So, the two triangles are congruent by the Side-Side-Side Congruence Property.

Due to the fact that ΔABC is congruent to ΔPQR and ΔPQR is a appropriate triangle, ΔABC need to also be a best triangle.

This is a contradiction. Thus, our assumption have to be wrong.


Example 1:

Check whether a triangle with side lengths 6 cm, 10 cm, and 8 cm is a ideal triangle.

Check whether the square of the size of the longest side is the amount of the squares of the various other 2 sides.

(10)2=?(8)2+(6)2  100=?64+36 100=100

Apply the converse of Pythagorean Theorem.

Because the square of the length of the longest side is the sum of the squares of the other 2 sides, by the converse of the Pythagorean Theorem, the triangle is a right triangle.

A corollary to the theorem categorizes triangles in to acute, appropriate, or obtusage.

In a triangle with side lengths a, b, and also c wright here c is the size of the longest side,

if c2a2+b2 then the triangle is acute, and

if c2>a2+b2 then the triangle is obtuse.


Example 2:

Check whether the triangle via the side lengths 5, 7, and also 9 devices is an acute, appropriate, or obtuse triangle.

The longest side of the triangle has a length of 9 devices.

Compare the square of the length of the longest side and the sum of squares of the various other two sides.

Square of the length of the longest side is 92=81 sq. units.

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Sum of the squares of the various other two sides is

52+72=25+49               =74sq.units

That is, 92>52+72.

Therefore, by the corollary to the converse of Pythagorean Theorem, the triangle is an obtuse triangle.