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The Pythagorean Theorem is a widely used mathematical theorem with many practical applications. The theorem states that in any right triangle, the sum of the squares of the two sides of the right angle is equal to the square of the hypotenuse. In other words, in a right triangle with right-angled sides of length a and b and hypotenuse c, we always have a 2 + b 2 = c 2 . The Pythagorean Theorem is one of the main pillars of basic geometry. There are many practical applications such as finding the distance between two points on a coordinate plane.
Steps
Find the sides of the right triangle
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Make sure your triangle is a right triangle. The Pythagorean theorem only applies to right triangles. So, before proceeding, make sure your triangle meets the criteria of a right triangle. Fortunately, there is only one criterion – to be a right triangle, the triangle must have an angle equal to 90 degrees.
- As a visual cue, the right angle is usually marked with a small square, which is not a circular “curve”. Look for this special mark at the corner of the triangle.
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Let the sides of the triangle be a, b, and c. In the Pythagorean Theorem, a and b are the sides of a right angle, and c is the hypotenuse – the longest side always opposite the right angle. So to start, call the shorter sides of the triangle a and b (it doesn’t matter which side is ‘a’ or ‘b’), and call the hypotenuse c.
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Determine which side of the triangle you need to find. The Pythagorean Theorem allows mathematicians to find the length of any one side of a right triangle as long as they know the lengths of the other two sides . Determine an edge of unknown length – a , b , and/or c . If only one edge is unknown, you can start.
- For example, suppose we know the hypotenuse has length 5 and one of the sides has length 3, but we don’t know what the length of the third side is. In this case, we will solve the problem of finding the third side, since the lengths of the other two sides are known. We will use this example in the next steps.
- If the length of two sides is unknown, you will need to determine the length of one more side to use the Pythagorean Theorem. Basic trigonometric functions can help you if you know the measure of one of the acute angles of a triangle.
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Substitute two known values into the equation. Substitute the lengths of the sides of the triangle into the equation a 2 + b 2 = c 2 . Remember that a and b are the two sides of the right angle, and c is the hypotenuse.
- In the above example, we know the lengths of one side and the hypotenuse (which are 3 and 5), so the equation would be 3² + b² = 5²
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Calculate squared. To solve the equation, start by taking the square of each known edge. Alternatively, if you find it easier, you can leave the lengths of the sides as exponents, and then square them later.
- In this example, we will square 3 and 5 to get 9 and 25 . The equation can be rewritten as 9 + b² = 25.
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Split the unknown variable to one side of the equation. If necessary, use basic algebra to move the unknown variable to one side of the equation and the two squared numbers to the other side of the equation. If you look for the hypotenuse, c is already on one side, so you don’t have to do anything to split it.
- In this example, the current equation is 9 + b² = 25. To separate b², subtract both sides of the equation by 9. The resulting equation is b² = 16.
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Take the square root of both sides of the equation. You will now be left with a squared variable on one side of the equation and a number on the other side. Simply take the square root of both sides to find the unknown side length.
- In this example, b² = 16, taking the square root of both sides we get b = 4. Thus, the length of the side to find is 4 .
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Use the Pythagorean Theorem to find the side of a right triangle in practice. The reason this Theorem is so widely used today is that it is applicable in a multitude of real-life situations. Learn to recognize right triangles in real life – any situation where two objects or two lines intersect at a right angle and a third object or line crosses through that right angle, you can use Determination Pythagorean theorem to find the length of one of the sides when the lengths of the other two are known.
- Take a real life example. A ladder is standing against the building. The bottom of the ladder is 5m from the bottom of the wall. The ladder is up to 20m high of the building. How many meters long is the ladder?
- The bottom of the ladder 5 m from the base of the wall and the height of 20 m of the building wall tells us the lengths of the two sides of the triangle. Since the wall and the ground intersect at a right angle and the ladder leans on the wall diagonally, we can think of it as a right triangle with side lengths a = 5 and b = 20. The ladder is hypotenuse, so c is unknown. Let’s use the Pythagorean Theorem:
- a² + b² = c²
- (5)² + (20)² = c²
- 25 + 400 = c²
- 425 = c²
- Square root of (425) = c
- c = 20.6. The length of the ladder is approximately 20.6 m.
- The bottom of the ladder 5 m from the base of the wall and the height of 20 m of the building wall tells us the lengths of the two sides of the triangle. Since the wall and the ground intersect at a right angle and the ladder leans on the wall diagonally, we can think of it as a right triangle with side lengths a = 5 and b = 20. The ladder is hypotenuse, so c is unknown. Let’s use the Pythagorean Theorem:
Calculate Distance between two points in XY . plane
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Identify two points in the XY plane. The Pythagorean Theorem can easily be used to calculate the straight line distance between two points in the XY plane. All you need to know are the x and y coordinates of any two points. Usually, these coordinates are written in coordinate pairs as (x, y).
- To find the distance between these two points, we will treat each point as one of the acute angles of a right triangle. This way, it’s easy to find the lengths of sides a and b, and then calculate side c or the distance between the two points.
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Draw two points on the graph. In the regular XY plane, for each point (x, y), x is the coordinate on the horizontal axis and y is the coordinate on the vertical axis. You can find the distance between two points without plotting them on a graph, but plotting will help you see better.
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Find the lengths of the right angles of the triangle. Using the two given points as the angles of the triangle adjacent to the hypotenuse, find the lengths of the sides a and b of the triangle. You can do this visually on the graph, or by using the formula |x 1 – x 2 | for horizontal edge and |y 1 – y 2 | for the vertical edge, where (x 1 ,y 1 ) is the first point and (x 2 ,y 2 ) is the second point.
- Let’s say the two points are (6,1) and (3,5). The length of the horizontal side of the triangle is:
- |x 1 – x 2 |
- |3 – 6|
- | -3 | = 3
- The length of the vertical side is:
- |y 1 – y 2 |
- |1 – 5|
- | -4 | = 4
- So, we can say that in this right triangle, side a = 3 and side b = 4.
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Use the Pythagorean Theorem to solve the equation for the hypotenuse. The distance between two given points is the hypotenuse of a triangle with two right angles as we just defined. Use the Pythagorean Theorem as usual to find the hypotenuse, let a be the length of the first side and b the length of the second.
- In the example with points (3,5) and (6,1), the lengths of the sides of the right angle are 3 and 4, so we calculate the length of the hypotenuse as follows:
-
- (3)²+(4)²= c²
- c= square root of (9+16)
- c= square root of (25)
- c= 5. The distance between two points (3,5) and (6,1) is 5 .
-
Advice
- The hypotenuse is always:
- cut across the right angle (do not go through the right angle)
- is the longest side of a right triangle
- represented by c in the Pythagorean theorem
- Always double check the results.
- Another way of checking – the longest side will face the largest angle and the shortest side will face the smallest angle.
- In a right triangle, you only know the third side when you know the lengths of the other two sides.
- If the triangle is not a right triangle, you will need more information beyond the lengths of the sides.
- To assign exact values to a, b, and c you should represent the triangle in graphical form, especially for logic or word problems.
- If you only have the measure of one side, you cannot use the Pythagorean Theorem. Instead use trigonometric functions (sin, cos, tan) or the ratio 30-60-90/45-45-90.
X
wikiHow is a “wiki” site, which means that many of the articles here are written by multiple authors. To create this article, 56 people, some of whom are anonymous, have edited and improved the article over time.
This article has been viewed 250,316 times.
The Pythagorean Theorem is a widely used mathematical theorem with many practical applications. The theorem states that in any right triangle, the sum of the squares of the two sides of the right angle is equal to the square of the hypotenuse. In other words, in a right triangle with right-angled sides of length a and b and hypotenuse c, we always have a 2 + b 2 = c 2 . The Pythagorean Theorem is one of the main pillars of basic geometry. There are many practical applications such as finding the distance between two points on a coordinate plane.
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