TLDR;
This video provides an introduction to the Pythagorean Theorem, explaining its application to right triangles and the relationship between their sides. It covers identifying the hypotenuse and legs of a right triangle, applying the formula a² + b² = c² to find missing side lengths, and a visual representation to solidify understanding.
- The Pythagorean Theorem applies only to right triangles.
- The formula a² + b² = c² is used to find missing side lengths, where 'c' is the hypotenuse and 'a' and 'b' are the legs.
- A visual representation demonstrates that the sum of the areas of the squares of the legs equals the area of the square of the hypotenuse.
Introduction to the Pythagorean Theorem [0:05]
The Pythagorean Theorem is introduced as a concept related to right triangles, named after the Greek mathematician Pythagoras. The theorem explains the relationship between the sides of any right triangle. The video emphasizes that the theorem applies exclusively to right triangles.
Identifying Hypotenuse and Legs [0:41]
Before applying the Pythagorean Theorem, it's crucial to identify the parts of a right triangle. The hypotenuse is the longest side, located directly opposite the right angle. The other two sides, which are shorter, are called legs. The Pythagorean Theorem states that the sum of the squares of the legs equals the square of the hypotenuse. This relationship is expressed as the equation a² + b² = c², where 'a' and 'b' represent the legs, and 'c' represents the hypotenuse. It is important to note that it doesn't matter which leg is assigned as 'a' or 'b'.
Example 1: Finding the Hypotenuse [2:39]
The video demonstrates how to use the Pythagorean Theorem to find a missing side length when two sides are known. Given a right triangle with legs of 4 feet and 3 feet, the goal is to find the length of the hypotenuse. By plugging the known values into the equation a² + b² = c², the equation becomes 4² + 3² = c². Solving this, 4² equals 16 and 3² equals 9, so the equation simplifies to 16 + 9 = c², which further simplifies to 25 = c². To isolate 'c', the square root of both sides is taken, resulting in c = 5. Therefore, the hypotenuse is 5 feet.
Visual Representation of the Pythagorean Theorem [5:24]
A visual representation is provided to aid in understanding the Pythagorean Theorem. Using the previous example of a right triangle with legs of 4 feet and 3 feet and a hypotenuse of 5 feet, squares are constructed on each side of the triangle. The areas of the squares on the legs (16 square feet and 9 square feet) add up to the area of the square on the hypotenuse (25 square feet), illustrating that a² + b² = c². Plugging in the values, 4² + 3² = 5², which simplifies to 16 + 9 = 25, confirming the theorem.
Example 2: Finding a Leg [9:04]
In this example, the video explains how to find the length of a missing leg when one leg and the hypotenuse are known. Given a right triangle with one leg measuring 15 cm and the hypotenuse measuring 17 cm, the task is to find the length of the other leg. Using the formula a² + b² = c², and substituting the known values, the equation becomes 15² + b² = 17². This simplifies to 225 + b² = 289. To isolate b², 225 is subtracted from both sides, resulting in b² = 64. Taking the square root of both sides isolates b, giving b = 8. Therefore, the missing leg is 8 cm.
