3rd Side Of A Triangle

The Pythagorean Theorem

Learning Objective(s)

· Use the Pythagorean Theorem to find the unknown side of a correct triangle.

· Solve awarding problems involving the Pythagorean Theorem.

Introduction

A long time ago, a Greek mathematician named Pythagoras discovered an interesting holding about correct triangles: the sum of the squares of the lengths of each of the triangle'due south legs is the same as the square of the length of the triangle'southward hypotenuse. This holding—which has many applications in science, fine art, technology, and compages—is now called the Pythagorean Theorem.

Let's take a look at how this theorem can help you learn more most the construction of triangles. And the all-time part—you lot 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.

The Pythagorean Theorem

If a and b are the lengths of the legs of a right triangle and c is the length of the hypotenuse, and so the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse.

This relationship is represented past the formula:

In the box above, y'all may have noticed the word "square," as well every bit the small 2s to the height right of the letters in . To foursquare a number ways to multiply information technology by itself. So, for example, to square the number five you multiply five • 5, and to square the number 12, you multiply 12 • 12. Some common squares are shown in the table below.

Number	Number Times Itself	Square
one	i^two = 1 • 1	one
2	2² = 2 • 2	four
three	iii² = 3 • 3	ix
4	4² = 4 • 4	16
5	5² = 5 • 5	25
x	10² = ten • 10	100

When yous meet the equation , you can think of this equally "the length of side a times itself, plus the length of side b times itself is the same every bit the length of side c times itself."

Permit's endeavour 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 foursquare of the length of the hypotenuse. And, in fact, it holds true for all right triangles.

The Pythagorean Theorem can as well be represented in terms of area. In any right triangle, the area of the foursquare drawn from the hypotenuse is equal to the sum of the areas of the squares that are fatigued from the ii legs. You can see this illustrated beneath in the aforementioned 3-iv-v right triangle.

Annotation that the Pythagorean Theorem only works with right triangles.

Finding the Length of the Hypotenuse

Yous can utilise the Pythagorean Theorem to find the length of the hypotenuse of a right triangle if you know the length of the triangle'due south other two sides, called the legs. Put another way, if you know the lengths of a and b, y'all can notice c.

In the triangle higher up, you are given measures for legs a and b: v and 12, respectively. You tin can utilise the Pythagorean Theorem to find a value for the length of c, the hypotenuse.

	_{The Pythagorean Theorem.}
	_{Substitute known values for a and b.}
	_Evaluate.
	_{Simplify. To find the value of c, remember about a number that, when multiplied by itself, equals 169. Does x work? How virtually 11? 12? thirteen? (You can employ a calculator to multiply if the numbers are unfamiliar.)}
_{13 = c}	_{The foursquare root of 169 is 13.}

Using the formula, you find that the length of c, the hypotenuse, is thirteen.

In this case, you did non know the value of c—yous were given the square of the length of the hypotenuse, and had to effigy it out from there. When you are given an equation similar and are asked to find the value of c, this is called finding the square root of a number. (Detect y'all plant a number, c, whose square was 169.)

Finding a square root takes some do, but it besides takes knowledge of multiplication, division, and a piffling bit of trial and fault. Expect at the table below.

Number x	Number y which, when multiplied by itself, equals number x	Square root y
1	1 • 1	ane
four	two • 2	2
9	3 • iii	3
16	4 • four	iv
25	v • 5	5
100	10 • ten	ten

It is a adept habit to become familiar with the squares of the numbers from 0‒10, as these arise oft in mathematics. If y'all can retrieve those square numbers—or if y'all can use a calculator to find them—then finding many mutual foursquare roots volition be just a thing of call up.

For which of these triangles is ?

Show/Hibernate Answer

Incorrect. This is not a right triangle, so you cannot use the Pythagorean Theorem to find r. The right answer is Triangle B.

Correct. This is a right triangle; when y'all sum the squares of the lengths of the sides, you get the foursquare of the length of the hypotenuse.

Incorrect. This is not a correct triangle, so yous cannot use the Pythagorean Theorem to find r. The correct respond is Triangle B.

Incorrect. This is not a right triangle, so yous cannot utilize the Pythagorean Theorem to find r. The correct reply is Triangle B.

Finding the Length of a Leg

Y'all can utilize the aforementioned formula to detect the length of a correct triangle's leg if y'all are given measurements for the lengths of the hypotenuse and the other leg. Consider the example beneath.

Example
Problem	Find the length of side a in the triangle below. Use a reckoner to estimate the foursquare root to one decimal place.
	a = ? b = 6 c = 7	In this right triangle, you are given the measurements for the hypotenuse, c, and one leg, b. The hypotenuse is always contrary the correct bending and it is always the longest side of the triangle.
		To notice the length of leg a, substitute the known values into the Pythagorean Theorem.
		Solve for a ². Call up: what number, when added to 36, gives you lot 49?
		Utilize a calculator to find the square root of 13. The calculator gives an answer of 3.6055…, which you can round to three.six. (Since y'all are approximating, you utilise the symbol .)
Reply

Which of the post-obit correctly uses the Pythagorean Theorem to find the missing side, ten?

B) ten + 8 = 10

Show/Hide Answer

Wrong. In this triangle, you know the hypotenuse (the side contrary the right bending) has a length of ten. The lengths of the legs are 8 and 10. The correct respond is .

B) 10 + 8 = ten

Wrong. The Pythagorean Theorem is a relationship between the lengths of the sides squared. The correct answer is .

Right. In this triangle, the hypotenuse has length 10, and the legs have length eight and x. Substituting into the Pythagorean Theorem yous have: ; this equation is the aforementioned as , or . What number, times itself, equals 36? That would make x = 6.

Incorrect. In this triangle, the hypotenuse has length 10 (always the longest side of the triangle and the side contrary the right angle) not 8. The correct answer is .

Using the Theorem to Solve Real Globe Bug

The Pythagorean Theorem is perhaps one of the most useful formulas yous 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.

Case
Problem	The owners of a business firm want to convert a stairway leading from the basis to their back porch into a ramp. The porch is 3 anxiety off the ground, and due to building regulations the ramp must offset 12 feet away from the base of the porch. How long volition the ramp exist? Use a figurer to find the foursquare root, and round the reply to the nearest tenth.
	To solve a problem like this one, it often makes sense to depict a simple diagram showing where the legs and hypotenuse of the triangle lie.

	a = 3 b = 12 c = ?	Identify the legs and the hypotenuse of the triangle. Yous know that the triangle is a correct triangle since the ground and the raised portion of the porch are perpendicular—this means you tin can use the Pythagorean Theorem to solve this problem. Identify a, b, and c.
		Employ the Pythagorean Theorem to find the length of c.
	12.four = c	Utilize a calculator to observe c. The square root of 153 is 12.369…, so you tin circular that to 12.four.
Respond	The ramp will be 12.four feet long.

Case
Problem	A sailboat has a large sail in the shape of a right triangle. The longest border of the canvass measures 17 yards, and the bottom edge of the sail is 8 yards. How tall is the sail?
		Describe an paradigm to help you lot visualize the problem. In a right triangle, the hypotenuse will always be the longest side, then hither it must be 17 yards. The trouble likewise tells you that the bottom edge of the triangle is 8 yards.
		Setup the Pythagorean Theorem.
	a = 15	15 • 15 = 225, then a = 15.
Respond	The elevation of the sheet is 15 yards.

Summary

The Pythagorean Theorem states that in whatever right triangle, the sum of the squares of the lengths of the triangle'due south legs is the same as the square of the length of the triangle'south 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.