Find B In Pythagorean Theorem

The Pythagorean theorem is a way of relating the leg lengths of a correct triangle to the length of the hypotenuse, which is the side opposite the right bending. Fifty-fifty though information technology is written in these terms, it tin be used to find any of the side as long every bit you know the lengths of the other two sides. In this lesson, we will look at several different types of examples of applying this theorem.

Table of Contents

Examples of using the Pythagorean theorem
Solving applied problems (word problems)
Solving algebraic problems
Summary

Applying the Pythagorean theorem (examples)

In the examples below, nosotros volition see how to apply this rule to find any side of a right triangle triangle. As in the formula below, we will let a and b be the lengths of the legs and c exist the length of the hypotenuse. Remember though, that yous could use whatsoever variables to represent these lengths.

image showing the pythagorean theorem formula

In each example, pay close attention to the information given and what we are trying to find. This helps yous make up one's mind the right values to utilize in the different parts of the formula.

Example

Find the value of \(ten\).

triangle with unknown hypotenuse

Solution

The side opposite the right angle is the side labelled \(10\). This is the hypotenuse. When applying the Pythagorean theorem, this squared is equal to the sum of the other two sides squared. Mathematically, this means:

\(half-dozen^two + 8^ii = 10^2\)

Which is the aforementioned as:

\(100 = x^two\)

Therefore, we can write:

\(\begin{marshal}ten &= \sqrt{100}\\ &= \bbox[edge: 1px solid black; padding: 2px]{ten}\terminate{align}\)

Maybe y'all recall that in an equation like this, \(10\) could also be –ten, since –10 squared is as well 100. But, the length of any side of a triangle can never be negative and therefore we only consider the positive foursquare root.

In other situations, yous will be trying to discover the length of one of the legs of a right triangle. You can still apply the Pythagorean theorem in these types of problems, but yous will need to be careful about the order you use the values in the formula.

Example

Discover the value of \(y\).

example where a leg length is unknown

Solution

The side opposite the right angle has a length of 12. Therefore, we will write:

\(8^2 + y^2 = 12^2\)

This is the aforementioned as:

\(64 + y^2 = 144\)

Subtracting 64 from both sides:

\(y^two = fourscore\)

Therefore:

\(\begin{marshal}y &= \sqrt{80} \\ &= \sqrt{16 \times 5} \\ &= \bbox[edge: 1px solid black; padding: 2px]{iv\sqrt{5}}\end{align}\)

In this terminal instance, we left the answer in exact form instead of finding a decimal approximation. This is mutual unless you are working on an practical problem.

Applications (word problems) with the Pythagorean theorem

There are many different kinds of existent-life issues that tin can be solved using the Pythagorean theorem. The easiest style to come across that you should be applying this theorem is by drawing a pic of any state of affairs is described.

Instance

Two hikers exit a motel at the same time, one heading due south and the other headed due west. After i hour, the hiker walking s has covered 2.eight miles and the hiker walking west has covered 3.one miles. At that moment, what is the shortest distance betwixt the 2 hikers?

Solution

First, sketch a picture of the data given. Label any unknown value with a variable proper name, like ten.

figure for a word problem with the pythagorean theorem

Due south and due due west class a right angle, and the shortest distance between any two points is a straight line. Therefore, nosotros tin can utilise the Pythagorean theorem and write:

\(3.1^two + ii.viii^2 = ten^ii\)

Here, yous will need to use a computer to simplify the left-hand side:

\(17.45 = x^two\)

At present employ your computer to accept the square root. You will likely need to circular your reply.

\(\begin{align}x &= \sqrt{17.45} \\ &\approx four.18 \text{ miles}\cease{align}\)

As you can encounter, it will be up to yous to decide that a right angle is part of the situation given in the give-and-take problem. If it isn't, then you lot can't use the Pythagorean theorem.

Algebra fashion problems with the Pythagorean theorem

There is one final type of problem yous might run into where you utilise the Pythagorean theorem to write some type of algebraic expression. This is something that you will not need to practise in every course, but it does come up.

Case

A right triangle has a hypotenuse of length \(2x\), a leg of length \(x\), and a leg of length y. Write an expression that shows the value of \(y\) in terms of \(x\).

Solution

Since no figure was given, your first step should be to describe one. The order of the legs isn't of import, but recollect that the hypotenuse is reverse the right bending.

algebra problem diagram

Now you tin can apply the Pythagorean theorem to write:

\(x^2 + y^two = (2x)^2\)

Squaring the correct-mitt side:

\(ten^2 + y^2 = 4x^ii\)

When the problem says "the value of \(y\)", it ways you must solve for \(y\). Therefore, we will write:

\(y^2 = 4x^2 – x^2\)

Combining like terms:

\(y^two = 3x^two\)

Now, use the foursquare root to write:

\(y = \sqrt{3x^2}\)

Finally, this simplifies to give the states the expression we are looking for:

\(y = \bbox[border: 1px solid black; padding: 2px]{x\sqrt{3x}}\)

Summary

The Pythagorean theorem allows yous to detect the length of whatever of the three sides of a right triangle. Information technology is one of those things that you should memorize, every bit it comes up in all areas of math, and therefore in many different math courses yous will probably accept. Retrieve to avoid the common mistake of mixing up where the legs go in the formula vs. the hypotenuse and to e'er draw a moving-picture show when ane isn't given.