Pythagoras Over 2000 years ago there was an amazing discovery about triangles: When a triangle has a right angle (90°) ... ... and squares are made on each of the three sides, ... geometry/images/pyth1.js ... then the biggest square has the exact same area as the other two squares put together! It is called "Pythagoras' Theorem" and can be written in one short equation: a2 + b2 = c2 Note:
DefinitionThe longest side of the triangle is called the "hypotenuse", so the formal definition is:
In a right angled triangle: the square of the hypotenuse is equal to the sum of the squares of the other two sides. Sure ... ?Let's see if it really works using an example.
Why Is This Useful?If we know the lengths of two sides of a right angled triangle, we can find the length of the third side. (But remember it only works on right angled triangles!) How Do I Use it?Write it down as an equation:
Start with:a2 + b2 = c2 Put in what we know:52 + 122 = c2 Calculate squares:25 + 144 = c2 25+144=169:169 = c2 Swap sides:c2 = 169 Square root of both sides:c = √169 Calculate:c = 13 Read Builder's Mathematics to see practical uses for this. Also read about Squares and Square Roots to find out why √169 = 13
Start with:a2 + b2 = c2 Put in what we know:92 + b2 = 152 Calculate squares:81 + b2 = 225 Take 81 from both sides: 81 − 81 + b2 = 225 − 81 Calculate: b2 = 144 Square root of both sides:b = √144 Calculate:b = 12
Start with:a2 + b2 = c2 Put in what we know:12 + 12 = c2 Calculate squares:1 + 1 = c2 1+1=2: 2 = c2 Swap sides: c2 = 2 Square root of both sides:c = √2 Which is about:c = 1.4142... It works the other way around, too: when the three sides of a triangle make a2 + b2 = c2, then the triangle is right angled.
Does a2 + b2 = c2 ?
They are equal, so ... Yes, it does have a Right Angle!
Does 82 + 152 = 162 ?
So, NO, it does not have a Right Angle
Does a2 + b2 = c2 ? Does (√3)2 + (√5)2 = (√8)2 ? Does 3 + 5 = 8 ? Yes, it does! So this is a right-angled triangle Get paper pen and scissors, then using the following animation as a guide:
Another, Amazingly Simple, ProofHere is one of the oldest proofs that the square on the long side has the same area as the other squares. Watch the animation, and pay attention when the triangles start sliding around. You may want to watch the animation a few times to understand what is happening. The purple triangle is the important one.
We also have a proof by adding up the areas. Historical Note: while we call it Pythagoras' Theorem, it was also known by Indian, Greek, Chinese and Babylonian mathematicians well before he lived. 511,512,617,618, 1145, 1146, 1147, 2359, 2360, 2361 Activity: Pythagoras' Theorem Copyright © 2022 Rod Pierce
We assume you're familiar with the Pythagorean Theorem. The converse of the Pythagorean Theorem is: If the square of the length of the longest side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle. That is, in ΔABC , if c 2 = a 2 + b 2 then ∠C is a right triangle, ΔPQR being the right angle. We can prove this by contradiction. Let us assume that c 2 = a 2 + b 2 in ΔABC and the triangle is not a right triangle. Now consider another triangle ΔPQR . We construct ΔPQR so that PR=a , QR=b and ∠R is a right angle. By the Pythagorean Theorem, ( PQ ) 2 = a 2 + b 2 . But we know that a 2 + b 2 = c 2 and a 2 + b 2 = c 2 and c=AB . So, ( PQ ) 2 = a 2 + b 2 = ( AB ) 2 . That is, ( PQ ) 2 = ( AB ) 2 . Since PQ and AB are lengths of sides, we can take positive square roots. PQ=AB That is, all the three sides of ΔPQR are congruent to the three sides of ΔABC . So, the two triangles are congruent by the Side-Side-Side Congruence Property. Since ΔABC is congruent to ΔPQR and ΔPQR is a right triangle, ΔABC must also be a right triangle. This is a contradiction. Therefore, our assumption must be wrong.
Example 1: Check whether a triangle with side lengths 6 cm, 10 cm, and 8 cm is a right triangle. Check whether the square of the length of the longest side is the sum of the squares of the other two sides. ( 10 ) 2 = ? ( 8 ) 2 + ( 6 ) 2 100 = ? 64+36 100=100 Apply the converse of Pythagorean Theorem. Since the square of the length of the longest side is the sum of the squares of the other two sides, by the converse of the Pythagorean Theorem, the triangle is a right triangle. A corollary to the theorem categorizes triangles in to acute, right, or obtuse. In a triangle with side lengths a , b , and c where c is the length of the longest side, if c 2 < a 2 + b 2 then the triangle is acute, and if c 2 > a 2 + b 2 then the triangle is obtuse.
Example 2: Check whether the triangle with the side lengths 5 , 7 , and 9 units is an acute, right, or obtuse triangle. The longest side of the triangle has a length of 9 units. Compare the square of the length of the longest side and the sum of squares of the other two sides. Square of the length of the longest side is 9 2 =81 sq. units. Sum of the squares of the other two sides is 5 2 + 7 2 =25+49 =74 sq. units That is, 9 2 > 5 2 + 7 2 . Therefore, by the corollary to the converse of Pythagorean Theorem, the triangle is an obtuse triangle. |
What kind of triangle is described if the square of the length of the longest side of a triangle is greater than the sum of the squares of the lengths of the other two sides?
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