Using The Distance Formula Or Pythagorean Theorem To Find The Distance Between Two Points

This tutorial examines how to find the distance between two points that are shown on the coordinate plane by drawing a right triangle and using the Pythagorean Theorem to solve

To find the hypotenuse of a right triangle with legs of length 6 and 8, we can use both the Pythagorean theorem and the distance formula.

1. **Using the Pythagorean Theorem:**
   The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

   Mathematically, it can be expressed as:
   \[ c^2 = a^2 + b^2 \]
   Where:
   - \( c \) is the length of the hypotenuse.
   - \( a \) and \( b \) are the lengths of the legs.

   Substituting the given lengths:
   \[ c^2 = 6^2 + 8^2 \]
   \[ c^2 = 36 + 64 \]
   \[ c^2 = 100 \]

   Taking the square root of both sides:
   \[ c = \sqrt{100} \]
   \[ c = 10 \]

   So, the length of the hypotenuse is \( 10 \) units.

2. **Using the Distance Formula:**
   The distance formula is derived from the Pythagorean theorem and is used to find the distance between two points in a coordinate plane.

   The distance formula between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
   \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

   In our case, we can consider the two points as the endpoints of the legs of the right triangle in a coordinate plane. Let's assume one leg is on the x-axis and the other on the y-axis. Then, the distance formula becomes:
   \[ c = \sqrt{(6 - 0)^2 + (0 - 8)^2} \]
   \[ c = \sqrt{6^2 + (-8)^2} \]
   \[ c = \sqrt{36 + 64} \]
   \[ c = \sqrt{100} \]
   \[ c = 10 \]

   Again, we find that the length of the hypotenuse is \( 10 \) units.

**Relation between the Distance Formula and the Pythagorean Theorem:**
The distance formula is essentially the application of the Pythagorean theorem in the context of coordinate geometry. It calculates the distance between two points in a plane by finding the length of the line segment connecting them, which is treated as the hypotenuse of a right triangle formed by the horizontal and vertical distances between the points. Thus, the distance formula and the Pythagorean theorem are closely related, with the distance formula being a specific application of the Pythagorean theorem in a coordinate system.

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