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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 Pythagoras Theorem (also called Pythagorean Theorem) is an important topic in Mathematics, which explains the relation between the sides of a right-angled triangle. The sides of the right triangle are also called Pythagorean triples. The formula and proof of this theorem are explained here with examples. Pythagoras theorem is basically used to find the length of an unknown side and the angle of a triangle. By this theorem, we can derive the base, perpendicular and hypotenuse formulas. Let us learn the mathematics of the Pythagorean theorem in detail here.
Pythagoras Theorem StatementPythagoras theorem states that “In a right-angled triangle, the square of the hypotenuse side is equal to the sum of squares of the other two sides“. The sides of this triangle have been named Perpendicular, Base and Hypotenuse. Here, the hypotenuse is the longest side, as it is opposite to the angle 90°. The sides of a right triangle (say a, b and c) which have positive integer values, when squared, are put into an equation, also called a Pythagorean triple. HistoryThe theorem is named after a Greek Mathematician called Pythagoras. Pythagoras Theorem FormulaConsider the triangle given above: Where “a” is the perpendicular, “b” is the base, “c” is the hypotenuse. According to the definition, the Pythagoras Theorem formula is given as:
The side opposite to the right angle (90°) is the longest side (known as Hypotenuse) because the side opposite to the greatest angle is the longest. Consider three squares of sides a, b, c mounted on the three sides of a triangle having the same sides as shown. By Pythagoras Theorem – Area of square “a” + Area of square “b” = Area of square “c” ExampleThe examples of theorem and based on the statement given for right triangles is given below: Consider a right triangle, given below: Find the value of x. X is the side opposite to the right angle, hence it is a hypotenuse. Now, by the theorem we know; Hypotenuse2 = Base2 + Perpendicular2 x2 = 82 + 62 x2 = 64+36 = 100 x = √100 = 10 Therefore, the value of x is 10. Pythagoras Theorem ProofGiven: A right-angled triangle ABC, right-angled at B. To Prove- AC2 = AB2 + BC2 Construction: Draw a perpendicular BD meeting AC at D. Proof: We know, △ADB ~ △ABC Therefore, (corresponding sides of similar triangles) Or, AB2 = AD × AC ……………………………..……..(1) Also, △BDC ~△ABC Therefore, (corresponding sides of similar triangles) Or, BC2= CD × AC ……………………………………..(2) Adding the equations (1) and (2) we get, AB2 + BC2 = AD × AC + CD × AC AB2 + BC2 = AC (AD + CD) Since, AD + CD = AC Therefore, AC2 = AB2 + BC2 Hence, the Pythagorean theorem is proved.
Note: Pythagorean theorem is only applicable to Right-Angled triangle. Video Lesson on Pythagoras TheoremApplications of Pythagoras Theorem
Useful ForPythagoras theorem is useful to find the sides of a right-angled triangle. If we know the two sides of a right triangle, then we can find the third side. How to use Pythagoras Theorem?To use Pythagoras theorem, remember the formula given below: c2 = a2 + b2 Where a, b and c are the sides of the right triangle. For example, if the sides of a triangles are a, b and c, such that a = 3 cm, b = 4 cm and c is the hypotenuse. Find the value of c. We know, c2 = a2 + b2 c2 = 32+42 c2 = 9+16 c2 = 25 c = √25 c = 5 cm Hence, the length of hypotenuse is 5 cm. How to find whether a triangle is a right-angled triangle?If we are provided with the length of three sides of a triangle, then to find whether the triangle is a right-angled triangle or not, we need to use the Pythagorean theorem. Let us understand this statement with the help of an example. Suppose a triangle with sides 10cm, 24cm, and 26cm are given. Clearly, 26 is the longest side. It also satisfies the condition, 10 + 24 > 26 We know, c2 = a2 + b2 ………(1) So, let a = 10, b = 24 and c = 26 First we will solve R.H.S. of equation 1. a2 + b2 = 102 + 242 = 100 + 576 = 676 Now, taking L.H.S, we get; c2 = 262 = 676 We can see, LHS = RHS Therefore, the given triangle is a right triangle, as it satisfies the Pythagoras theorem. Related ArticlesPythagorean Theorem Solved ExamplesProblem 1: The sides of a triangle are 5, 12 & 13 units. Check if it has a right angle or not. Solution: From Pythagoras Theorem, we have; Perpendicular2 + Base2 = Hypotenuse2 P2 + B2 = H2 Let, Perpendicular (P) = 12 units Base (B)= 5 units Hypotenuse (H) = 13 units {since it is the longest side measure} LHS = P2 + B2 ⇒ 122 + 52 ⇒ 144 + 25 ⇒ 169 RHS = H2 ⇒ 132 ⇒ 169 ⇒ 169 = 169 L.H.S. = R.H.S. Therefore, the angle opposite to the 13 units side will be a right angle. Problem 2: The two sides of a right-angled triangle are given as shown in the figure. Find the third side. Perpendicular = 15 cm Base = b cm Hypotenuse = 17 cm As per the Pythagorean Theorem, we have; Perpendicular2 + Base2 = Hypotenuse2 ⇒152 + b2 = 172 ⇒225 + b2 = 289 ⇒b2 = 289 – 225 ⇒b2 = 64 ⇒b = √64 Therefore, b = 8 cm Problem 3: Given the side of a square to be 4 cm. Find the length of the diagonal. Solution- Given; Sides of a square = 4 cm To Find- The length of diagonal ac. Consider triangle abc (or can also be acd) (ab)2 +(bc)2 = (ac)2 (4)2 +(4)2= (ac)2 16 + 16 = (ac)2 32 = (ac)2 (ac)2 = 32 ac = 4√2. Thus, the length of the diagonal is 4√2 cm. Stay tuned with BYJU’S – The Learning App to learn all the important mathematical concepts and also watch interactive videos to learn with ease.
The formula for Pythagoras, for a right-angled triangle, is given by; P2 + B2 = H2
Pythagoras theorem states that, in a right triangle, the square of the hypotenuse is equal to the sum of the square of the other two sides.
The hypotenuse is the longest side of the right-angled triangle, opposite to right angle, which is adjacent to base and perpendicular. Let base, perpendicular and hypotenuse be a, b and c respectively. Then the hypotenuse formula, from the Pythagoras statement will be;
No, this theorem is applicable only for the right-angled triangle. The theorem can be used to find the steepness of the hills or mountains. To find the distance between the observer and a point on the ground from the tower or a building above which the observer is viewing the point. It is mostly used in the field of construction.
Yes, the diagonals of a square can be found using the Pythagoras theorem, as the diagonal divides the square into right triangles.
Step 1: To find the unknown sides of a right triangle, plug the known values in the Pythagoras theorem formula. Step 2: Simplify the equation to find the unknown side. Step 3: Solve the equation for the unknown side.
There are various approaches to prove the Pythagoras theorem. A few of them are listed below: Proof using similar triangles Proof using differentials Euclid’s proof Algebraic proof, and so on. |
In what kind of triangle does the square of the longest side is equal equal to the sum of the squares of the other two remaining sides?
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