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The Pythagorean Theorem is a fundamental concept in mathematics that allows us to calculate the length of an unknown side in a right triangle. It is named after the ancient Greek mathematician Pythagoras, who discovered this mathematical relationship between the lengths of the sides of a right triangle. In this guide, we will learn how to use the Pythagorean Theorem to solve various problems involving right triangles. Whether you are a student learning geometry or someone who needs to make measurements in everyday life, understanding and utilizing the Pythagorean Theorem is invaluable. So, let’s dive into the world of right triangles and explore how we can apply this theorem to solve practical problems.
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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
- 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.
- 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.
- 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
- In this example, we will square 3 and 5 to get 9 and 25 . The equation can be rewritten as 9 + b² = 25.
- 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.
- 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 .
- 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
- 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.
- 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.
- 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.
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.
In conclusion, the Pythagorean Theorem is a fundamental concept in mathematics that allows us to find the length of a side in a right triangle. We can use this theorem to solve a wide range of problems, including finding the distance between two points on a coordinate plane, determining the height of buildings or objects, and calculating the lengths of sides in various geometric shapes. By understanding the Pythagorean Theorem and practicing its application, we can enhance our problem-solving skills and gain a deeper understanding of geometry. It is a powerful tool that has practical applications in numerous fields, including construction, architecture, engineering, and physics. Whether you are a student, professional, or simply interested in mathematics, mastering the Pythagorean Theorem is essential for tackling geometric challenges and expanding your mathematical knowledge.
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