Scientists and mathematicians like to study relationships. They study how a plant or animal relates to its environment, and how numbers relate to nature and music. When a specific relationship is found over and over, they give it a name and share the information. One example is the Pythagorean Theorem, which explains the relationship between the sides of a right triangle.
The Anatomy of a Triangle
Before you can learn about the Pythagorean Theorem, you need to know a few terms.
A right triangle has one ninety-degree angle inside the triangle, which is called a right angle. Mathematicians often mark the right angle with a box. On the diagram of the triangle, the box is drawn in red.
In a right triangle, the hypotenuse is the side directly across from the right angle. It is the only side of the triangle that is not a part of the right angle. On the diagram, the hypotenuse is green.
An exponent appears slightly above and to the right of a number, like this: 32. A number with an exponent is actually a multiplication sentence, and the exponent indicates how many times that number appears in the multiplication sentence. So this example, 32, means 3 x 3. An exponent of 2 is read ''to the second power'' or ''squared.''
The Pythagorean Theorem
The Pythagorean Theorem states the relationship between the sides of a right triangle, when c stands for the hypotenuse and a and b are the sides forming the right angle. The formula is:
a2 + b2 = c2
It is read ''a-squared plus b-squared equals c-squared.''
Using the Pythagorean Theorem
Leila's school is having a door-decorating contest. Leila's class wants to use ribbon to divide their door into two triangles. They know the length of each side of the door. To find out how long the ribbon needs to be, they can use the Pythagorean Theorem.
Look at the picture of the door. The door is a rectangle, so the corner is a right angle. The right angle is drawn in blue. The sides forming the right angle are labeled a and b. The ribbon is drawn in blue, dividing the door into two right triangles. The ribbon is opposite the right angle, which means it is the hypotenuse. It is labeled c. You can see that a is 4 feet and b is 3 feet, so we can use the Pythagorean Theorem to find c.
a2 + b2 = c2
Plug In the Numbers
42 + 32 = c2
Solve the Exponents
- 42 means 4 x 4, which equals 16
- 32 means 3 x 3, which equals 9
- The new equation is 16 + 9 = c2
Solve for c
- 16 + 9 = c2
- 25 = c2
- Now we need to solve for c. Since we learned that c x c = 25, we can use multiplication facts to realize that 5 x 5 = 25. c = 5. The ribbon needs to be 5 feet long.
The Pythagorean Theorem describes the relationships between the sides of a right triangle. The square of the hypotenuse, the side opposite the right angle, is equal to the sum of the squares of the two sides. The formula is a2 + b2 = c2.
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Real-World Applications of the Pythagorean Theorem:
- The Pythagorean Theorem applies to right triangles, where a right triangle is a triangle with a right, or 90-degree, angle.
- The hypotenuse of a right triangle is the side opposite the 90-degree angle in the triangle.
- The Pythagorean Theorem states that if a right triangle has two sides with lengths a and b, and a hypotenuse of length c, then a^2 + b^2 = c^2.
- Suppose a support beam is to be built so that it runs between the floor and a wall. The distance up the wall that the beam reaches is 8 feet, and the other end of the beam is on the floor 6 feet from the wall. What should the length of the support beam be?
- A surveyor is trying to determine the distance across a pond without having to actually get in the pond to measure it. Because of how the pond is shaped, they are able to create a right triangle with a hypotenuse being the distance across the pond, and the other two sides are on land. The surveyor measures the two sides on land to be 24 feet and 32 feet. What is the distance across the pond?
Solutions and Explanations:
- In this scenario, the beam, the floor, and the wall create a right triangle, with the beam being the hypotenuse, and the other two sides having lengths 6 feet and 8 feet. Thus, the Pythagorean Theorem gives that 6^2 + 8^2 = c^2, where c is the length of the support beam. Simplifying the left-hand side of this equation gives 100 = c^2. Since 10 times 10 is 100, we know that c = 10, so the support beam should be 10 feet.
- In this problem, the distance across the pond is the hypotenuse of a right triangle with sides of length 24 feet and 32 feet, so the Pythagorean Theorem gives that 24^2 + 32^2 = c^2, with c being the distance across the pond. Simplifying the left-hand side of this equation gives 1,600 = c^2. Since 40 times 40 equals 1,600, we have that c = 40. Therefore, the distance across the pond is 40 feet.
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