Video transcriptSay we have a right triangle. Let me draw my right triangle just like that. This is a right triangle. This is the 90 degree angle right here. And we're told that this side's length right here is 14. This side's length right over here is 9. And we're told that this side is a. And we need to find the length of a. So as I mentioned already, this is a right triangle. And we know that if we have a right triangle, if we know two of the sides, we can always figure out a third side using the Pythagorean theorem. And what the Pythagorean theorem tells us is that the sum of the squares of the shorter sides is going to be equal to the square of the longer side, or the square of the hypotenuse. And if you're not sure about that, you're probably thinking, hey Sal, how do I know that a is shorter than this side over here? How do I know it's not 15 or 16? And the way to tell is that the longest side in a right triangle, and this only applies to a right triangle, is the side opposite the 90 degree angle. And in this case, 14 is opposite the 90 degrees. This 90 degree angle kind of opens into this longest side. The side that we call the hypotenuse. So now that we know that that's the longest side, let me color code it. So this is the longest side. This is one of the shorter sides. And this is the other of the shorter sides. The Pythagorean theorem tells us that the sum of the squares of the shorter sides, so a squared plus 9 squared is going to be equal to 14 squared. And it's really important that you realize that it's not 9 squared plus 14 squared is going to be equal to a squared. a squared is one of the shorter sides. The sum of the squares of these two sides are going to be equal to 14 squared, the hypotenuse squared. And from here, we just have to solve for a. So we get a squared plus 81 is equal to 14 squared. In case we don't know what that is, let's just multiply it out. 14 times 14. 4 times 4 is 16. 4 times 1 is 4 plus 1 is 5. Take a 0 there. 1 times 4 is 4. 1 times 1 is 1. 6 plus 0 is 6. 5 plus 4 is 9, bring down the 1. It's 196. So a squared plus 81 is equal to 14 squared, which is 196. Then we could subtract 81 from both sides of this equation. On the left-hand side, we're going to be left with just the a squared. These two guys cancel out, the whole point of subtracting 81. So we're left with a squared is equal to 196 minus 81. What is that? If you just subtract 1, it's 195. If you subtract 80, it would be 115 if I'm doing that right. And then to solve for a, we just take the square root of both sides, the principal square root, the positive square root of both sides of this equation. So let's do that. Because we're dealing with distances, you can't have a negative square root, or a negative distance here. And we get a is equal to the square root of 115. Let's see if we can break down 115 any further. So let's see. It's clearly divisible by 5. If you factor it out, it's 5, and then 5 goes in the 115 23 times. So both of these are prime numbers. So we're done. So you actually can't factor this anymore. So a is just going to be equal to the square root of 115. Now if you want to get a sense of roughly how large the square root of 115 is, if you think about it, the square root of 100 is equal to 10. And the square root of 121 is equal to 11. So this value right here is going to be someplace in between 10 and 11, which makes sense if you think about it visually. Show
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a2 + b2 = c2, where a and b are the two shorter sides of a right-angled triangle and c is the longest side, opposite the right angle. The theorem is used to find a missing side of a right-angled triangle when the other two sides are known. Pythagoras theorem is used to determine a missing side of a right-angled triangle when 2 other sides are known. To use Pythagoras theorem, the triangle must contain a right-angle and 2 side lengths must be known. To use Pythagoras Theorem:
For example, find the missing hypotenuse of this triangleThis triangle has two shorter sides of length 5 cm and 9 cm. The two shorter sides are labelled as a and b. It does not matter which is a or which is b. c is labelled as the hypotenuse. In other words, c is always the longest side of a right-angled triangle.  Pythagoras theorem is a2 + b2 = c2. This can be rearranged for c by taking the square root of both sides of the equation. The formula for a missing hypotenuse becomes: First square the two shorter sides of 5 and 9 to get a2 = 25 and b2 = 81. Next we add a2 and b2 together. 25 + 81 = 106. Finally we square root this result. √106 = 10.3 to 1 decimal place. To check the Pythagorean theorem calculation, the calculated hypotenuse should be larger than the two shorter sides. 10.3 is larger than both 5 and 9 but not by a very large magnitude. It is therefore less likely that a mistake has been made and our result is more likely to be correct. For example, find the missing shorter side of this triangleThis triangle has a hypotenuse of 11 cm and one of the shorter sides is 6 cm. We need to find the other shorter side on the base of the triangle. The hypotenuse, which is the side opposite to the right angle, is always labelled as c. Label a as the missing shorter side that you wish to find. Label b as the shorter side that you know the length of. Pythagoras theorem is a2 + b2 = c2. This can be rearranged for a shorter side, a by subtractingb2 from both sides of the equation to get a2 = c2 - b2. Taking the square root of both sides, the formula for a missing shorter side becomes: We first square both known sides. c = 11 and b = 6. Therefore, c2 = 121 and b2 = 36. To work out c2 - b2, we work out 121 - 36. This equals 85. Finally, we square root this to find the missing side. √85 = 9.2 to 1 decimal place. How to Find the Hypotenuse Using PythagorasThe hypotenuse is the longest side of a right-angled triangle. To find it:
How to Find the Shorter Side Using PythagorasTo find one of the shorter sides using Pythagoras:
Pythagoras’ Theorem CalculatorTo use Pythagoras theorem, you must know the length of two sides of a right-angled triangle. Enter the two known sides in the solver below and it will automatically calculate the missing side. Simple Proof of Pythagoras’ TheoremTo prove Pythagoras theorem:
Pythagoras theorem always works for any right-angled triangle because it has been proven algebraically. Several algebraic proofs exist linking the squares of each side of a right-angled triangle. Step 1. Place 4 triangles with sides a, b and c to form a square where each side is made by combining sides a and b We take four of these identical triangles and place them so that side a and side b are next to each other, forming a square. We can also see a diagonal inner square has formed with side lengths of c. Step 2. Find the area of the outer square Each side of the outer square is made from sides a and b combined. Therefore the length of the outer square is (a + b). Therefore the area of the outer square is (a + b)2. (a + b)2 = (a + b)(a + b) = a2 + 2ab + b2. Step 3. Subtract the area of the 4 triangles Each of the blue triangles has base = b and height = a. The area of a triangle is . The area of each triangle is therefore . Since we have four triangles, the total area of all four triangles is . This simplifies to 2ab. Step 4. Subtract the area of the 4 triangles from the area of the outer square to leave the area of the inner square The area of the outer square subtract the area of the four triangles equals the area of the inner square.
The Converse of Pythagoras’ TheoremThe converse of Pythagoras theorem states that a triangle is right-angled only if its sides obey Pythagoras theorem. That is, if the two smaller sides squared sum to the same value as the largest side squared, the triangle contains a right angle. Pythagoras theorem only works for right-angled triangles. It also must always work if we have a right-angled triangle. Therefore Pythagoras theorem is a perfect method to use to test if a triangle contains a right-angle or not. Pythagoras theorem is often used to find a missing side, when we already know the triangle contains a right angle. The converse of Pythagoras theorem is when we already know all three sides but use these to check if there is a right angle in the first place. The converse of Pythagoras theorem is used as a test to decide if a triangle contains a right angle or not. All three sides of the triangle must be known to do the test. For example, the triangle containing side lengths of 3, 4 and 5 does contain a right angle. This is because 32 + 42 = 52. 9 + 16 = 25 and so the triangle satisfies Pythagoras theorem and so it must contain a right angle. For example, the triangle containing side lengths of 2, 4 and 6 does not contain a right angle. This is because 22 + 42 ≠ 62. 4 + 16 ≠ 36 and so the triangle cannot contain a right angle. How Do You Know When To Use Pythagoras or Trigonometry?If you need to find a missing angle, you must use trigonometry. If you need to find a missing side, Pythagoras or trigonometry can be used. If you have two known sides in a triangle, use Pythagoras to find the third side. If you know a side and an angle, use trigonometry. Both Pythagoras theorem and trigonometry can be used with right-angled triangles. To decide whether to use Pythagoras theorem or trigonometry, use the following rules:
For example, find the missing side on this triangle To find a missing side, Pythagoras or trigonometry can be used. Since we know the other 2 side lengths, we know that we use Pythagoras. We would need to have known an angle to use trigonometry. b = 10 and c = 20 Therefore, b2 = 100 and c2 = 400 Since we are finding a shorter side, we subtract these and then square root the answer. and so, . The missing side, a = 17.3 For example, find the missing side on this triangle To find a missing side, Pythagoras or trigonometry can be used. Since we know one side and know an angle, we use trigonometry. We would need to know the 2 other sides to use Pythagoras. We want the opposite side and we know the hypotenuse. Using SOHCAHTOA, we will use the SOH ratio since we have H and need O. O = H × sin(θ) and so, O = 10 × sin(30) = 5. Therefore the missing side length is 5. Pythagoras’ Theorem in 3DPythagoras theorem can be used in 3D problems involving right-angled triangles. Pythagoras equation in 3D tells us that the length of the diagonal of a cuboid is d = √(𝑥2 + y2 + z2), where𝑥, y and z are the side lengths of the cuboid. The diagonal of a cuboid is , where 𝑥, y and z are the side lengths of the cuboid. We can derive Pythagoras theorem in 3D by considering 2 separate right-angled triangles within a cuboid. We can do Pythagoras in 2D in the pink triangle for the base triangle to find the diagonal base length, c. We can then use this side length in the green triangle to calculate the diagonal length, d. For example, use Pythagoras theorem to find the diagonal length of a cube with side lengths of 5.Pythagoras theorem in 3D is , where 𝑥, y and z are the side lengths of the cuboid. In a cube, all sides are the same length and so, 𝑥, y and z are all equal to 5. The equation becomes, which equals 8.66. Pythagorean TriplesPythagorean triples are three integers a, b and c, such that a2 + b2 = c2. For example 3, 4 and 5 form the a Pythagorean triple because 32 + 42 = 52. Pythagorean triples can be found using the formulae of a = m2 - n2, b = 2mn, c = m2 + n2. List of Pythagorean TriplesHere is a complete list of the first Pythagorean triples:
There are an infinite number of Pythagorean triples. Some Pythagorean triples are multiples of smaller Pythagorean triples. For example if we multiply (3, 4, 5) by 2, we get another triple: (6, 8, 10). If a Pythagorean triple is not a multiple of another smaller Pythagorean triple, then it is called a primitive Pythagorean Triple. For example, (3, 4, 5) and (5, 12, 13) are primitive Pythagorean triples, however, (10, 24, 26) is not a primitive Pythagorean triple because is it 2 × (5, 12, 13). To find primitive Pythagorean triples, substitute integer values of m and n into the formulae:
For example, if m = 2 and n = 1, the first Pythagorean triple is found:
The first Pythagorean triple is (3, 4, 5). We can then find all other Pythagorean triples by multiplying these primitive Pythagorean triples by any integer number. Properties of Primitive Pythagorean TriplesThe following properties apply to primitive Pythagorean triples:
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Use pythagorean theorem to find the missing side
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