How to Use the Pythagorean Theorem: 12 Steps (with Pictures)

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How to Use the Pythagorean Theorem: 12 Steps (with Pictures)
How to Use the Pythagorean Theorem: 12 Steps (with Pictures)

Video: How to Use the Pythagorean Theorem: 12 Steps (with Pictures)

Video: How to Use the Pythagorean Theorem: 12 Steps (with Pictures)
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The Pythagorean theorem describes the lengths of the sides of a right triangle in an elegant and practical way, so this theorem is still widely used today. This theorem states that for any right triangle, the sum of the squares of the non-angled sides is equal to the square of the hypotenuse. In other words, for a right triangle with perpendicular sides a and b and hypotenuse c, a2 + b2 = c2.

The Pythagorean theorem is one of the basic pillars of elementary geometry. There are countless applications using this theorem, for example, to make it easy to find the distance between two points on a coordinate plane.

Step

Method 1 of 2: Finding the Sides of a Right Triangle

Use the Pythagorean Theorem Step 1
Use the Pythagorean Theorem Step 1

Step 1. Make sure that your triangle is a right triangle

The Pythagorean theorem only applies to right triangles, so, before proceeding, it is very important to make sure that your triangles conform to the properties of right triangles. Fortunately, there is one factor that can indicate that your triangle is a right triangle. Your triangle should have one 90 degree angle.

As a sign, right triangles are often marked with small squares to mark 90-degree angles, not using curved "curves." Look for this particular mark in the corner of your triangle

Use the Pythagorean Theorem Step 2
Use the Pythagorean Theorem Step 2

Step 2. Give variables a, b, and c for the sides of your triangle

In the Pythagorean Theorem, the variables a and b represent the sides that meet at the right triangle, while the variable c represents the hypotenuse – the long side opposite the right angle. So, to begin with, mark the short sides of your triangle with the variables a and b (it doesn't matter if you swap them), and mark the hypotenuse with the variable c.

Use the Pythagorean Theorem Step 3
Use the Pythagorean Theorem Step 3

Step 3. Decide which side of the triangle you want to solve

The Pythagorean theorem allows mathematicians to find the length of any side of a right triangle as long as they know the lengths of the other two sides. Determine which side is unknown - a, b, and/or c. If the length of one of your sides is unknown, you are ready to move on.

  • For example, we know that the length of the hypotenuse of a triangle is 5 and the length of one of the other sides is 3, but we are not sure of the length of the third side. In this case, we know that we are looking for the length of the third side, and since we know the lengths of the other two, we can solve it! We will work on this problem with the following steps.
  • If you don't know the lengths of two sides, you must know one of the sides to be able to use the Pythagorean Theorem. Basic trigonometric functions can help you if you know one side of a triangle that is not slanted.
Use the Pythagorean Theorem Step 4
Use the Pythagorean Theorem Step 4

Step 4. Plug the two-sided values you already know into the equation

Plug the lengths of the sides of your triangle into the equation a2 + b2 = c2. Remember that a and b are non-sloping sides, while c is the hypotenuse.

In our example, we know the length of one of the sides and the hypotenuse (3 & 5), so the equation becomes 3² + b² = 5²

Use the Pythagorean Theorem Step 5
Use the Pythagorean Theorem Step 5

Step 5. Square

To solve your equation, start by squaring the known sides. Alternatively, if you find this easier, you can leave your side lengths squared, and square them later.

  • In our example, we will square 3 and 5 so that we get

    Step 9. da

    Step 25.. We can write the equation as 9 + b² = 25.

Use the Pythagorean Theorem Step 6
Use the Pythagorean Theorem Step 6

Step 6. Move the unknown variable to the other side of the equation

If needed, use basic algebraic operations to make the unknown variable move to the other side of the equation and the square of the other two variables to the other side. If you want to find the length of the hypotenuse, c is already on the other side of the equation, so you don't have to do anything to move it.

In our example, the current equation is 9 + b² = 25. To move b², subtract both sides of the equation by 9, so the result is b² = 16

Use the Pythagorean Theorem Step 7
Use the Pythagorean Theorem Step 7

Step 7. Square root of both sides of the equation

Now only one variable is squared on one side and number on the other. Square root of both sides to find the length of the unknown side.

  • In our example, b² = 16, taking the square root of both sides gives b = 4. Thus, we can say that the length of the unknown side of the triangle is

    Step 4..

Use the Pythagorean Theorem Step 8
Use the Pythagorean Theorem Step 8

Step 8. Use the Pythagorean Theorem to find the sides of a true right triangle

The reason the Pythagorean Theorem is widely used today is that it can be applied to countless practical situations. Learn to know right triangles in real life – in any situation where two objects or straight lines meet a right angle and the third object or line joins the two objects or lines diagonally, then you can use the Pythagorean Theorem to find the length of the side the other, if the lengths of the other two sides are known.

  • Let's try a real-life example which is a bit more difficult. A ladder leans against a building. The distance from the bottom of the stairs to the wall is 5 meters. The height of the stairs reaches 20 meters. How long is the ladder?

    • 5 meters from the wall and 20 meters high tells us the lengths of the sides of the triangle. Since the wall and the ground (assumed) form a right angle and the ladder is propped diagonally against the wall, this arrangement can be considered a right triangle with side lengths a = 5 and b = 20. The length of the ladder is the hypotenuse, so the value of c is not is known. Let's use the Pythagorean Theorem:

      • a² + b² = c²
      • (5)² + (20)² = c²
      • 25 + 400 = c²
      • 425 = c²
      • root(425) = c
      • c = 20.6. The approximate length of the ladder is 20.6 meters.

Method 2 of 2: Calculating the Distance Between Two Points in the X-Y. Plane

Use the Pythagorean Theorem Step 9
Use the Pythagorean Theorem Step 9

Step 1. Find two points in the X-Y plane

The Pythagorean theorem can be used easily to calculate the straight line distance between two points in the X-Y plane. All you need to know is the x and y coordinates of the two points. Usually, these coordinates are written together in the form (x, y).

To find the distance between these two points, we will consider each point as one of the non-right angles of a right triangle. Doing so will make it easy to find the lengths of sides a and b, and then calculate the hypotenuse c, which is the distance between the two points

Use the Pythagorean Theorem Step 10
Use the Pythagorean Theorem Step 10

Step 2. Draw your two points in the image

In a regular X-Y plane, each point (x, y), x represents a horizontal coordinate and y represents a vertical coordinate. You can find the distance between the two points without drawing it, but doing so will give you a visual image that you can use to see if your answer is correct.

Use the Pythagorean Theorem Step 11
Use the Pythagorean Theorem Step 11

Step 3. Find the length of the non-sloping side of your triangle

Using the two 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 using an image or using the formula |x1 - x2| for the horizontal side and |y1 - y2| for the vertical side, with (x1, y1) as the first point and (x2, y2) as the second point.

  • Let our two points be (6, 1) and (3, 5). The length of the horizontal side of our triangle is:

    • |x1 - x2|
    • |3 - 6|

    • | -3 | =

      Step 3.

  • The length of the vertical side is:

    • |y1 - y2|
    • |1 - 5|

    • | -4 | =

      Step 4.

  • So, in our right triangle, side a = 3 and side b = 4.
Use the Pythagorean Theorem Step 12
Use the Pythagorean Theorem Step 12

Step 4. Use the Pythagorean Theorem to find the length of the hypotenuse

The distance between two points is the length of the hypotenuse of the triangle whose two sides you just found. Use the Pythagorean Theorem to find the hypotenuse, where a is the length of the first side and b is the length of the second side.

  • In our example, we are using the points (3, 5) and (6, 1) whose side lengths are 3 and 4, so we can find the hypotenuse as follows:

    • (3)²+(4)²= c²
      c= root (9+16)
      c= root(25)

      c= 5. The distance between (3, 5) and (6, 1) is

      Step 5..

Tips

  • The hypotenuse is always:

    • opposite the right angle (without touching the right angle)
    • longest side in a right triangle
    • called c in the Pythagorean theorem
  • root (x) means the square root of x.
  • Remember to always check your answers. If your answer seems wrong, try again and try again.
  • If the triangle is not a right triangle, you need additional information, not just the lengths of the other two sides.
  • Another way of checking – the longest side is opposite the largest angle and the shortest side is opposite the smallest angle.
  • Figures are the key to writing down the correct values for a, b, and c. If you are working on a story problem, be sure to write down the problem in picture form first.
  • If you only know the length of one side, the Pythagorean Theorem doesn't work. Try using trigonometry (sin, cos, tan) or 30-60-90 / 45-45-90 ratios.

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