Content

• Steps
• Method 1 of 2: Finding the Sides of a Right Triangle
• Method 2 of 2: Calculating the Distance Between Two Points on a Coordinate Plane
• Tips

The Pythagorean theorem connects the three sides of a right-angled triangle with one formula, which is still used today. The theorem says that in a right-angled triangle, the sum of the squares of the legs is equal to the square of the hypotenuse: a + b = c, where a and b are the legs of the triangle (sides intersecting at right angles), c is the hypotenuse of the triangle. The Pythagorean theorem is applicable in many cases, for example, using this theorem, it is easy to find the distance between two points on the coordinate plane.

Steps

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

1. 1 Make sure the triangle you are given is right-angled, as the Pythagorean theorem only applies to right-angled triangles. In right-angled triangles, one of the three angles is always 90 degrees.
   • A right angle in a right triangle is indicated by a square icon, not a curve, which is an oblique angle.
2. 2 Add guidelines for the sides of the triangle. Label the legs as "a" and "b" (legs - sides intersecting at right angles), and the hypotenuse as "c" (hypotenuse - the largest side of a right triangle lying opposite a right angle).
3. 3 Determine which side of the triangle you want to find. The Pythagorean theorem allows you to find any side of a right triangle (if the other two sides are known). Determine which side (a, b, c) you need to find.
   • For example, given a hypotenuse equal to 5, and given a leg equal to 3. In this case, you need to find the second leg. We'll come back to this example later.
   • If the other two sides are unknown, it is necessary to find the length of one of the unknown sides in order to be able to apply the Pythagorean theorem. To do this, use the basic trigonometric functions (if you are given the value of one of the oblique angles).
4. 4 Substitute the values ​​you give (or the values ​​you find) into the formula a + b = c. Remember that a and b are legs and c is hypotenuse.
   • In our example, write: 3² + b² = 5².
5. 5 Square each side you know. Or leave the degrees - you can square the numbers later.
   • In our example, write: 9 + b² = 25.
6. 6 Isolate the unknown side on one side of the equation. To do this, transfer the known values ​​to the other side of the equation. If you find the hypotenuse, then in the Pythagorean theorem it is already isolated on one side of the equation (so nothing needs to be done).
   • In our example, move 9 to the right side of the equation to isolate the unknown b². You will get b² = 16.
7. 7 Take the square root of both sides of the equation. At this stage, there is an unknown (squared) on one side of the equation, and an intercept (number) on the other side.
   • In our example, b² = 16. Take the square root of both sides of the equation and get b = 4. So the second leg is 4.
8. 8 Use the Pythagorean theorem in your daily life, as it can be applied in a wide variety of practical situations. To do this, learn to recognize right-angled triangles in everyday life - in any situation in which two objects (or lines) intersect at right angles, and a third object (or line) connects (diagonally) the tops of the first two objects (or lines), you can use the Pythagorean theorem to find the unknown side (if the other two sides are known).
   • Example: given a staircase leaning against a building. The bottom of the stairs is 5 meters from the base of the wall. The top of the stairs is 20 meters from the ground (up the wall). How long are the stairs?
     • “5 meters from the base of the wall” means that a = 5; "Located 20 meters from the ground" means that b = 20 (that is, you are given two legs of a right-angled triangle, since the wall of the building and the surface of the Earth intersect at right angles). The length of the ladder is the length of the hypotenuse, which is unknown.
       • a² + b² = c²
       • (5) ² + (20) ² = c²
       • 25 + 400 = c²
       • 425 = c²
       • c = √425
       • s = 20.6. So the approximate length of the ladder is 20.6 meters.

Method 2 of 2: Calculating the Distance Between Two Points on a Coordinate Plane

1. 1 Select two points on the coordinate plane. By the Pythagorean theorem, you can calculate the length of the segment connecting two points on the coordinate line.To do this, you need to know the coordinates (x, y) of each point.
   • To find the distance between two points, you will consider the points as the vertices of a triangle, not adjacent to the right angle of a right triangle. Thus, you can easily find the legs of the triangle, and then calculate the hypotenuse, which is equal to the distance between two points.
2. 2 Draw points on the coordinate plane. Set aside the coordinates (x, y), where the x coordinate is along the horizontal axis and the y coordinate along the vertical. You can find the distance between points without drawing a graph, but a graph allows you to visually represent the process of your calculations.
3. 3 Find the legs of the triangle. You can do this by measuring the length of the legs directly on the graph or using the formulas: | x₁ - x₂| to calculate the length of the horizontal leg, and | y₁ - y₂| to calculate the length of the vertical leg, where (x₁, y₁) Are the coordinates of the first point, and (x₂, y₂) - coordinates of the second point.
   • Example: given points: A (6.1) and B (3.5). Horizontal leg length:
     • | x₁ - x₂|
     • |3 - 6|
     • | -3 | = 3
   • Length of the vertical leg:
     • | y₁ - y₂|
     • |1 - 5|
     • | -4 | = 4
   • Thus, in a right-angled triangle, a = 3 and b = 4.
4. 4 Use the Pythagorean theorem to find the hypotenuse. The distance between two points is equal to the hypotenuse of the triangle, the two sides of which you just found. Use the Pythagorean theorem to find the hypotenuse by substituting the found values ​​of the legs (a and b) into the formula.
   • In our example, a = 3 and b = 4. The hypotenuse is calculated as follows:

     • (3) ² + (4) ² = c²

       c = √ (9 + 16)

       c = √ (25)

       c = 5. The distance between points A (6.1) and B (3.5) is 5.

Tips

• Hypotenuse is always:
  • lies opposite a right angle;
  • is the longest side of a right-angled triangle;
  • denoted as "c" in the Pythagorean theorem;
• √ (x) means "square root of x".
• Don't forget to check the answer. If the answer seems wrong, do the calculations again.
• Another point is that the longest side is opposite the largest corner, and the shortest side is opposite the smallest corner.
• Learn the numbers of the Pythagorean triplet that form the sides of a right triangle. The most primitive Pythagorean triplet is 3, 4, 5. So, knowing the length of two sides, you don't have to look for a third.
  • Remember, the hypotenuse is always the longest side.
• If you are given a regular triangle (rather than a rectangular one), then more information is required than just the lengths of the two sides.
• Graphs are a visual way of drawing designations a, b and c. If you are solving a problem, build a graph first.
• If the length of only one side is given, then the Pythagorean theorem cannot be applied. Try using trigonometry (sin, cos, tan).
• If we are talking about a problem from a certain plot, we can safely assume that trees, pillars, walls, and so on form a right angle with the ground, unless otherwise indicated.