As a form of visual shorthand, right angles are often marked with a small square, rather than a rounded “curve”, to identify them as such. Look for this special mark in one of the corners of your triangle.
Let’s say, for example, that we know that our hypotenuse has a length of 5 and one of the other sides has a length of 3, but we’re not sure what the length of the third side is. In this case, we know we’re solving for the length of the third side, and, because we know the lengths of the other two, we’re ready to go! We’ll return to this example problem in the following steps. If the lengths of two of your sides are unknown, you’ll need to determine the length of one more side to use the Pythagorean Theorem. Basic trigonometry functions can help you here if you know one of the non-right angles in the triangle.
In our example, we know the length of one side and the hypotenuse (3 & 5), so we would write our equation as 3² + b² = 5²
In our example, we would square 3 and 5 to get 9 and 25, respectively. We can rewrite our equation as 9 + b² = 25.
In our example, our current equation is 9 + b² = 25. To isolate b², let’s subtract 9 from both sides of the equation. This leaves us with b² = 16.
In our example, b² = 16, taking the square root of both sides gives us b = 4. Thus, we can say that the length of the unknown side of our triangle is 4.
Let’s try real-world example that’s a little more difficult. A ladder is leaning against a building. The base of the ladder is 5 meters (16. 4 ft) from the bottom of the wall. The ladder reaches 20 meters (65. 6 ft) up the wall of the building. How long is the ladder? “5 meters (16. 4 ft) from the bottom of wall” and “20 meters (65. 6 ft) up the wall” clue us into the lengths of the sides of our triangle. Since the wall and the ground (presumably) meet at a right angle and the ladder leans diagonally against the wall, we can think of this arrangement as a right triangle with sides of length a = 5 and b = 20. The length of the ladder is the hypotenuse, so c is our unknown. Let’s use the Pythagorean Theorem: a² + b² = c² (5)² + (20)² = c² 25 + 400 = c² 425 = c² sqrt(425) = c c = 20. 6 . The approximate length of the ladder is 20. 6 meters (67. 6 ft).
To find the distance between these two points, we will treat each point as one of the non-right angle corners of a right triangle. By doing this, it’s easy to find the length of the a and b sides, then calculate c, the hypotenuse, which is the distance between the two points.
Let’s say our two points are (6,1) and (3,5). The side length of the horizontal side of our triangle is: |x1 - x2| |3 - 6| | -3 | = 3 The length of the vertical side is: |y1 - y2| |1 - 5| | -4 | = 4 So, we can say that in our right triangle, side a = 3 and side b = 4.
In our example using points (3,5) and (6,1), our side lengths are 3 and 4, so we would find the hypotenuse as follows: (3)²+(4)²= c² c= sqrt(9+16) c= sqrt(25) c= 5. The distance between (3,5) and (6,1) is 5.
