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Math 102: College Mathematics14 chapters | 108 lessons
DaQuita has taught high school mathematics for six years and has a master's degree in secondary mathematics education.
Jessica has a square garden. To make watering her plants easier, she would like to create a diagonal pathway that connects the opposite corners. If one side of her garden is 13 feet long, what will be the length of her diagonal pathway?
This scenario presents a perfect example of geometry in real life. Since the garden is in the shape of a square, we will be able to use the Pythagorean theorem to figure out how long the diagonal pathway will be. Let's take a closer look at this theorem and how it's used so that we can calculate the solution to this problem.
Written by the Greek mathematician Pythagoras, the Pythagorean theorem states that in right triangles, the sum of the squares of the two legs is equal to the square of the hypotenuse. In clearer terms, if you square both of the legs and then add these numbers together, you will end up with the same value as the hypotenuse squared. As a formula, this theorem is written as a^2 + b^2 = c^2, where a and b represent the two legs of the triangle, and c represents the hypotenuse.
When labeling any right triangle with these variables, it is important to know that a and b can be written on either leg. However, c must be written on the hypotenuse, which is the longest side and opposite the right angle.
The purpose of the Pythagorean theorem is to make it possible to find the length of any missing side of a right triangle. For the theorem to be usable, the value of the other two sides must be known.
Let's take a look at a few basic problems.
In our first triangle, one leg has a length of 4 centimeters, and the second leg has a length of 7 centimeters. Let's find the length of the hypotenuse. Since I have the value for the two legs, which are represented by a and b, we will let a = 4 and b = 7.
We are now ready to plug this information into the formula. Once we do, we get 4^2 + 7^2 = c^2. After simplifying, we end up with 16 + 49 = c^2, and when we combine our like terms, we have 65 = c^2. For our last step, we will take the square root of both sides to get a final answer of c = 8.06. Therefore, we can see that the length of the diagonal is 8.06 centimeters.
In our next triangle, the measure of one leg is 12 inches and the measure of the hypotenuse is 20 inches. For this triangle, we have the value of the hypotenuse, but must find the length of the missing leg. Therefore, we will let a = 12 and c = 20.
By substituting this information into the formula, we have 12^2 + b^2 = 20^2. When we simplify, we get 144 + b^2 = 400. At this point, we should subtract 144 from both sides to get b^2 = 256. Once we take the square root of both sides, we see that b, the missing leg in our triangle, has a value of 16 inches.
Did anything about the numbers in that last example stand out to you? Notice that all of the side lengths are whole numbers, not fractions or decimals. This means that the side lengths for this triangle are a Pythagorean triple, which is defined as a set of nonzero whole numbers that satisfy the Pythagorean theorem.
A few more examples of Pythagorean triples are 3, 4, 5, and 28, 45, 53. If you plug these numbers into the Pythagorean theorem, you will see that 3^2 + 4^2 = 5^2, and 28^2 + 45^2 = 53^2.
Now that we have an understanding of the Pythagorean theorem, let's apply it to a few everyday examples, beginning with Jessica's garden from the beginning of the lesson.
Let's review the problem again. Jessica has a square garden. To make watering her plants easier, she would like to create a diagonal pathway that connects opposite corners. If one side of her garden is 13 feet long, what will be the length of her diagonal pathway?
For this situation, we should definitely use the Pythagorean theorem. We know that in a square, all four angles measure 90 degrees and all four sides are congruent. So if one side has a length of 13 feet, then all the other sides have a length of 13 feet. By taking half of this square, the diagonal becomes the hypotenuse and we have a right triangle with both legs equal to 13 feet.
We are now ready to plug in our information to find the length of the hypotenuse. When we do, we will see that 13^2 + 13^2 = c^2. By simplifying and combining like terms, we end up with 338 = c^2, and c = 18.38 as our final answer, which will be the length of Jessica's diagonal pathway.
Let's do another. Elizabeth needs to repair her roof. She props a 21-foot ladder against the side of her 20-foot home. If the top of the ladder reaches the top of her house, how far away from her home is the base of the ladder?
In this problem, we are still working with a right triangle. The ladder is the hypotenuse, while the height of the home and the ground between the home and the ladder are the legs of this right triangle.
Using the given information, our beginning equation is 20^2 + b^2 = 21^2. Simplifying brings us to 400 + b^2 = 441. From here, we subtract 400 from both sides to get b^2 = 41, from which we see that b = 6.40. Therefore, we can conclude that the base of the ladder is 6.40 feet away from the home.
In summary, the Pythagorean theorem, which can only be used with right triangles, is a very important theorem that allows us to find the length of a missing side in a triangle. The theorem is interpreted as the formula a^2 + b^2 = c^2. The legs are represented by a and b and can be labeled interchangeably, but the hypotenuse must be labeled c. In order for this theorem to be used, two sides of the right triangle must be known.
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