One of the best known mathematical formulas is Pythagorean Theorem, which provides us with the relationship between the sides in a right triangle. A right triangle consists of two legs and a hypotenuse. The two legs meet at a 90° angle and the hypotenuse is the longest side of the right triangle and is the side opposite the right angle. The Pythagorean Theorem tells us that the relationship in every right triangle is: $$a^{2}+b^{2}=c^{2}$$ Example $$C^{2}=6^{2}+4^{2}$$ $$C^{2}=36+16$$ $$C^{2}=52$$ $$C=\sqrt{52}$$ $$C\approx 7.2$$ There are a couple of special types of right triangles, like the 45°-45° right triangles and the 30°-60° right triangle. Because of their angles it is easier to find the hypotenuse or the legs in these right triangles than in all other right triangles. In a 45°-45° right triangle we only need to multiply one leg by √2 to get the length of the hypotenuse. Example We multiply the length of the leg which is 7 inches by √2 to get the length of the hypotenuse. $$7\cdot \sqrt{2}\approx 9.9$$ In a 30°-60° right triangle we can find the length of the leg that is opposite the 30° angle by using this formula: $$a=\frac{1}{2}\cdot c$$ Example To find a, we use the formula above. $$a=\frac{1}{2}\cdot 14$$ $$a=7$$ Video lessonFind the sides of this right triangle The legs of a right triangle are the sides that are adjacent to its right angle. Sometimes we have problems that ask us to find a missing length of one of these legs. We can use the Pythagorean theorem to find a missing leg of a triangle, but only if we know the length measure of the hypotenuse and the other one of the legs. The Pythagorean Theorem only applies to a right triangle. And in that right triangle we can say that the legs a and b that is those sides that are adjacent to this right angle. If I square them and add them up it's going to equal your hypotenuse squared or c squared so your hypotenuse remember is that side that is opposite your 90 degree angle and since the triangle angle sum says that if this is 90 degrees then both of these angles have to be less than 90, your
hypotenuse will always be the longest side in a right triangle. The Pythagorean Theorem Learning Objective(s) · Use the Pythagorean Theorem to find the unknown side of a right triangle. · Solve application problems involving the Pythagorean Theorem. Introduction A long time ago, a Greek mathematician named Pythagoras discovered an interesting property about right triangles: the sum of the squares of the lengths of each of the triangle’s legs is the same as the square of the length of the triangle’s hypotenuse. This property—which has many applications in science, art, engineering, and architecture—is now called the Pythagorean Theorem. Let’s take a look at how this theorem can help you learn more about the construction of triangles. And the best part—you don’t even have to speak Greek to apply Pythagoras’ discovery. The Pythagorean Theorem Pythagoras studied right triangles, and the relationships between the legs and the hypotenuse of a right triangle, before deriving his theory.
In the box above, you may have noticed the word “square,” as well as the small 2s to the top right of the letters in . To square a number means to multiply it by itself. So, for example, to square the number 5 you multiply 5 • 5, and to square the number 12, you multiply 12 • 12. Some common squares are shown in the table below.
When you see the equation , you can think of this as “the length of side a times itself, plus the length of side b times itself is the same as the length of side c times itself.” Let’s try out all of the Pythagorean Theorem with an actual right triangle. This theorem holds true for this right triangle—the sum of the squares of the lengths of both legs is the same as the square of the length of the hypotenuse. And, in fact, it holds true for all right triangles. The Pythagorean Theorem can also be represented in terms of area. In any right triangle, the area of the square drawn from the hypotenuse is equal to the sum of the areas of the squares that are drawn from the two legs. You can see this illustrated below in the same 3-4-5 right triangle. Note that the Pythagorean Theorem only works with right triangles. Finding the Length of the Hypotenuse You can use the Pythagorean Theorem to find the length of the hypotenuse of a right triangle if you know the length of the triangle’s other two sides, called the legs. Put another way, if you know the lengths of a and b, you can find c. In the triangle above, you are given measures for legs a and b: 5 and 12, respectively. You can use the Pythagorean Theorem to find a value for the length of c, the hypotenuse.
Using the formula, you find that the length of c, the hypotenuse, is 13. In this case, you did not know the value of c—you were given the square of the length of the hypotenuse, and had to figure it out from there. When you are given an equation like and are asked to find the value of c, this is called finding the square root of a number. (Notice you found a number, c, whose square was 169.) Finding a square root takes some practice, but it also takes knowledge of multiplication, division, and a little bit of trial and error. Look at the table below.
It is a good habit to become familiar with the squares of the numbers from 0‒10, as these arise frequently in mathematics. If you can remember those square numbers—or if you can use a calculator to find them—then finding many common square roots will be just a matter of recall. Finding the Length of a Leg You can use the same formula to find the length of a right triangle’s leg if you are given measurements for the lengths of the hypotenuse and the other leg. Consider the example below.
Which of the following correctly uses the Pythagorean Theorem to find the missing side, x? A) B) x + 8 = 10 C) D) Using the Theorem to Solve Real World Problems The Pythagorean Theorem is perhaps one of the most useful formulas you will learn in mathematics because there are so many applications of it in real world settings. Architects and engineers use this formula extensively when building ramps, bridges, and buildings. Look at the following examples.
Summary The Pythagorean Theorem states that in any right triangle, the sum of the squares of the lengths of the triangle’s legs is the same as the square of the length of the triangle’s hypotenuse. This theorem is represented by the formula . Put simply, if you know the lengths of two sides of a right triangle, you can apply the Pythagorean Theorem to find the length of the third side. Remember, this theorem only works for right triangles. |