Ratios in right triangles

Let’s refresh some concepts that will come in handy as you start the right triangles and trigonometry unit of the high school geometry course. You’ll see a summary of each concept, along with a sample item, links for more practice, and some info about why you will need the concept for the unit ahead.

This article only includes concepts from earlier courses. There are also concepts within this high school geometry course that are important to understanding right triangles and trigonometry. If you have not yet mastered the Introduction to triangle similarity lesson, it may be helpful for you to review that before going farther into the unit ahead.

##Pythagorean theorem

###What is this, and why do we need it?

The Pythagorean theorem is $a^2+b^2=c^2$, where $a$ and $b$ are lengths of the legs of a right triangle and $c$ is the length of the hypotenuse. The theorem means that if we know the lengths of any two sides of a right triangle, we can find out the length of the last side. We can find right triangles all over the place—inside of prisms and pyramids, on maps when we're finding distance, even hiding inside of equilateral triangles!

###Practice

[[☃ graded-group-set 1]]

For more practice, go to Use Pythagorean theorem to find right triangle side lengths.

###Where will we use this?

Here are a few of the exercises where reviewing the Pythagorean theorem might be helpful:

• Pythagorean theorem in 3D
• Pythagorean theorem challenge
• Special right triangles

##Simplify square root expressions

###What is this, and why do we need it?

For geometry, the square root function takes the area of a square as the input and give the length of a side of the square as an output. We'll use square root expressions when we use the Pythagorean theorem to find a side length. The trigonometric ratios for benchmark angles like $30\degree$, $45\degree$, and $60\degree$ depend on square root expressions.

###Practice

[[☃ graded-group-set 3]]

For more practice, go to Simplify square roots and Simplify square root expressions.

###Where will we use this?

Here are a few of the exercises where reviewing square root expressions might be helpful.

• Pythagorean theorem in 3D
• Pythagorean theorem challenge
• Special right triangles

##Visualizing right triangles in context

###What is this, and why do we need it?

Remember how there are right triangles hiding everywhere? To apply the Pythagorean theorem and trigonometry in context, we need to notice where the right angles are and think about what the hypotenuse and legs represent. Then we figure out where the measurements we have fit into the picture.

###Practice

[[☃ graded-group-set 2]]

We don't have an exercise for this, because the best way to practice is by drawing your own diagrams on paper or your surface of choice!

###Where will we use this?

Here are a few of the exercises where practicing visualizing right triangles might be helpful:

• Pythagorean theorem in 3D
• Pythagorean theorem challenge
• Right triangle trigonometry word problems

By the end of the unit, you should be able to find all of the unlabeled lengths and angle measurements in the diagrams, not just the ones we asked about. Come back at the end of the unit and see how much you've learned!

We use special words to describe the sides of right triangles.

The hypotenuse of a right triangle is always the side opposite the right angle. It is the longest side in a right triangle.

[[☃ image 1]]

The other two sides are called the opposite and adjacent sides. These sides are labeled in relation to an angle.

The opposite side is across from a given angle.

[[☃ image 2]]

The adjacent side is the non-hypotenuse side that is next to a given angle.

[[☃ image 3]]

Putting it all together from the perspective of $\angle A$:

[[☃ image 4]]

And from $\angle B$:

[[☃ image 5]]

Practice

[[☃ graded-group-set 1]]

Why are these words important?

We're about to learn the trigonometric functions—sine, cosine, and tangent—which are defined using the words hypotenuse, opposite, and adjacent.

When we studied congruence, we claimed that knowing two angle measures and the side length between them (Angle-Side-Angle congruence) was enough for being sure that *all* of the corresponding pairs of sides and angles were congruent.

[[☃ image 1]]

How can that be? Even with the Pythagorean theorem, we need two side lengths to find the third. In this article, we'll take the first steps towards understanding how the angle measures and side lengths give us information about each other in the special case of right triangles.

This is a great opportunity to work with a friend or two. The goal of this article is to find and discuss patterns, not to spend a bunch of time calculating. Try splitting up the work so there's more time to talk about what you see!

##Let's look for patterns

First, we'll collect some data about a set of triangles.

[[☃ graded-group 1]]

[[☃ graded-group 2]]

[[☃ explanation 1]]

Now we're ready to start checking that data for patterns.

[[☃ graded-group 3]]

[[☃ explanation 2]]

**What did you notice?**

[[☃ explanation 3]]

##Proving that the pattern works for another angle measure

[[☃ graded-group 4]]

[[☃ graded-group 5]]

##What did we conclude?

If two right triangles have an acute angle measure in common, they are similar by angle-angle similarity. The ratios of corresponding side lengths *within* the triangles will be equal. So the ratio of the side lengths of a right triangle just depends on one acute angle measure.

##Why will this be useful?

Before, we could use the Pythagorean theorem to find any missing side length of a right triangle when we knew the other two lengths. Now, we have a way to relate angle measures to the right triangle side lengths. That allows us to find both missing side lengths when we only know one length and an acute angle measure. We can even find the acute angle measures in a right triangle based on any two side lengths.

[[☃ graded-group-set 1]]

What are the trigonometric ratios?

The ratios of the sides of a right triangle are called trigonometric ratios. Three common trigonometric ratios are the sine (sin), cosine (cos), and tangent (tan). These are defined for acute angle $A$ below:

[[☃ image 1]]

In these definitions, the terms opposite, adjacent, and hypotenuse refer to the lengths of the sides.

Learn more with our Trigonometric ratios in right triangles video.

Practice with our Trigonometric ratios in right triangles exercise.

Where are these topics used in the real world?

Trigonometry is used in a lot of different fields! Architects and engineers use trigonometry to design buildings and bridges. Surveyors use it to measure distances and angles. Astronomers use trigonometry to measure distances between stars and galaxies. In addition, carpenters, artists, and even athletes can use the principles of right triangle trigonometry in their work.

What do we know about the sine and cosine of complementary angles?

Knowing the sine and cosine of complementary angles can be helpful when solving problems with right triangles. The sine of an angle is equal to the cosine of its complementary angle, and vice versa. So if we know the cosine of an angle, we can use that information to find the sine of its complementary angle, or vice versa.

Practice with our Relate ratios in right triangles exercise.

How do we use the reciprocal trigonometric ratios?

The reciprocal trigonometric ratios are just the inverses of the regular trigonometric ratios: cosecant is the reciprocal of sine, secant is the reciprocal of cosine, and cotangent is the reciprocal of tangent. We can use them in the same way we use the regular trigonometric ratios, to solve for side lengths or angles in right triangles.

Practice with our Reciprocal trig ratios
exercise.

What do we mean by modeling with right triangles?

We can use right triangles to model real-world situations. For example, we might use a right triangle to figure out the height of a building or the distance across a river. Modeling with right triangles can help us solve problems we wouldn't be able to solve otherwise.

Practice with our Right triangle trigonometry word problems
exercise.