To find the geometric mean of these two segments, multiply the lengths and find the square root of the product.
So, the geometric mean of the two segments would be a line segment that is in length. Use these concepts and strategies to complete example 2.
Example 2
In below, what is the geometric mean of and ?
When finding a geometric mean, you first find the product of the items involved. In this case, segment is and segment is . Then find the square root of this product.
So, the geometric mean of and in is .
Altitude as Geometric Mean
In a right triangle, the length of the altitude from the right angle to the hypotenuse is the geometric mean of the lengths of the two segments of the hypotenuse. In the diagram below we can use to create the proportion . Solving for .
You can use this relationship to find the length of the altitude if you know the length of the two segments of the divided hypotenuse.
Example 3
What is the length of the altitude in the triangle below?
To find the altitude of this triangle, find the geometric mean of the two segments of the hypotenuse. In this case, you need to find the geometric mean of and . To find the geometric mean, find the product of the two numbers and then take its square root.
So , or approximately .
Example 4
What is the length of the altitude in the triangle below?
The altitude of this triangle is . Remember the altitude does not always go “down”! To find , find the geometric mean of the two segments of the hypotenuse. Make sure that you fill in missing information in the diagram. You know that the whole hypotenuse, is long and , but you need to know , the length of the longer subsection of , to find the geometric mean. To do this, subtract.
So . Write this measurement on the diagram to keep track of your work.
Now find the geometric mean of and to identify the length of the altitude.
The altitude of the triangle will measure .
Leg as Geometric Mean
Just as we used similar triangles to create a proportion using the altitude, the lengths of the legs in right triangles can also be found with a geometric mean with respect to the hypotenuse. The length of one leg in a right triangle is the geometric mean of the adjacent segment and the entire hypotenuse. The diagram below shows the relationships.
You can use this relationship to find the length of the leg if you know the length of the two segments of the divided hypotenuse.
Example 5
What is the length of in the triangle below?
To find , the leg of the large right triangle, find the geometric mean of the adjacent segments of the hypotenuse and the entire hypotenuse. In this case, you need to find the geometric mean of and . To find the geometric mean, find the product of the two numbers and then take the square root of that product.
So, or approximately .
Example 6
If , what is the value in the triangle below?
To find in this triangle, find the geometric mean of the adjacent segment of the hypotenuse and the entire hypotenuse. Make sure that you fill in missing information in the diagram. You know that the two shorter sections of the hypotenuse are and , but you need to know the length of the entire hypotenuse to find the geometric mean. To do this, add.
So, . Write this measurement on the diagram to keep track of your work.
Now find the geometric mean of and to identify the length of the altitude.
So, .
Lesson Summary
In this lesson, we explored how to work with different radicals both in theory and in practical situations. Specifically, we have learned:
How to identify similar triangles inscribed in a larger triangle.
How to evaluate the geometric mean of various objects.
How to identify the length of an altitude using the geometric mean of a separated hypotenuse.
How to identify the length of a leg using the geometric mean of a separated hypotenuse.
These skills will help you solve many different types of problems. Always be on the lookout for new and interesting ways to find relationships between sides and angles in triangles.
Points to Consider
How can you use the Pythagorean Theorem to identify other relationships between sides in triangles?
Review Questions
Which triangles in the diagram below are similar?
What is the geometric mean of two line segments that are and , respectively?
What is the geometric mean of two line segments that are each?
Which triangles in the diagram below are similar?
What is the length of the altitude, , in the triangle below?
What is the length of in the triangle below?
What is the geometric mean of two line segments that are and , respectively?
What is the length of the altitude in the triangle below?
Use the following diagram from exercises 9-11:
____
____
____ (for an extra challenge, find in two different ways)
What is the length of the altitude in the triangle below?
Review Answers
Triangles and are all similar
Triangles and are all similar.
, or approximately
, or approximately
, or approximately
, or approximately
or approximately
or approximately . One way to find is with the geometric mean: . Alternatively, using the answer from 9 and one of the smaller right triangles,
.
Special Right Triangles
Learning Objectives
Identify and use the ratios involved with right isosceles triangles.
Identify and use the ratios involved with triangles.
Identify and use ratios involved with equilateral triangles.
Employ right triangle ratios when solving real-world problems.
Introduction
What happens when you cut an equilateral triangle in half using an altitude? You get two right triangles. What about a square? If you draw a diagonal across a square you also get two right triangles. These two right triangles are special special right triangles called the and the right triangles. They have unique properties and if you understand the relationships between the sides and angles in these triangles, you will do well in geometry, trigonometry, and beyond.
Right Isosceles Triangles
The first type of right triangle to examine is isosceles. As you know, isosceles triangles have two sides that are the same length. Additionally, the base angles of an isosceles triangle are congruent as well. An isosceles right triangle will always have base angles that each measure and a vertex angle that measures .
Don’t forget that the base angles are the angles across from the congruent sides. They don’t have to be on the bottom of the figure.
Because the angles of all triangles will, by definition, remain the same, all triangles are similar, so their sides will always be proportional. To find the relationship between the sides, use the Pythagorean Theorem.
Example 1
The isosceles right triangle below has legs measuring .
Use the Pythagorean Theorem to find the length of the hypotenuse.
Since the legs are each, substitute for both and , and solve for :
In this example .
What if each leg in the example above was ? Then we would have
If each leg is , then the hypotenuse is .
When the length of each leg was , the hypotenuse was . When the length of each leg was , the hypotenuse was . Is this a coincidence? No. Recall that the legs of all triangles are proportional. The hypotenuse of an isosceles right triangle will always equal the product of the length of one leg and . Use this information to solve the problem in example 2.
Example 2
What is the length of the hypotenuse in the triangle below?
Si
nce the length of the hypotenuse is the product of one leg and , you can easily calculate this length. One leg is , so the hypotenuse will be , or about .
Equilateral Triangles
Remember that an equilateral triangle has sides that all have the same length. Equilateral triangles are also equiangular—all angles have the same measure. In an equilateral triangle, all angles measure exactly .
Notice what happens when you divide an equilateral triangle in half.
When an equilateral triangle is divided into two equal parts using an altitude, each resulting right triangle is a triangle. The hypotenuse of the resulting triangle was the side of the original, and the shorter leg is half of an original side. This is why the hypotenuse is always twice the length of the shorter leg in a triangle. You can use this information to solve problems about equilateral triangles.
30º-60º-90º Triangles
Another important type of right triangle has angles measuring , , and . Just as you found a constant ratio between the sides of an isosceles right triangle, you can find constant ratios here as well. Use the Pythagorean Theorem to discover these important relationships.
Example 3
Find the length of the missing leg in the following triangle. Use the Pythagorean Theorem to find your answer.
Just like you did for triangles, use the Pythagorean theorem to find the missing side. In this diagram, you are given two measurements: the hypotenuse is and the shorter leg is . Find the length of the missing leg .
You can leave the answer in radical form as shown, or use your calculator to find the approximate value of .
On your own, try this again using a hypotenuse of . Recall that since the triangle comes from an equilateral triangle, you know that the length of the shorter leg is half the length of the hypotenuse.
Now you should be able to identify the constant ratios in triangles. The hypotenuse will always be twice the length of the shorter leg, and the longer leg is always the product of the length of the shorter leg and . In ratio form, the sides, in order from shortest to longest are in the ratio .
Example 4
What is the length of the missing leg in the triangle below?
Since the length of the longer leg is the product of the shorter leg and , you can easily calculate this length. The short leg is , so the longer leg will be , or about .
Example 5
What is below?
To find the length of segment , identify its relationship to the rest of the triangle. Since it is an altitude, it forms two congruent triangles with angles measuring , , and . So, will be the product of (the shorter leg) and .
, or approximately .
Special Right Triangles in the Real World
You can use special right triangles in many real-world contexts. Many real-life applications of geometry rely on special right triangles, so being able to recall and use these ratios is a way to save time when solving problems.
Example 6
The diagram below shows the shadow a flagpole casts at a certain time of day.
If the length of the shadow cast by the flagpole is , what is the height of the flagpole and the length of the hypotenuse of the right triangle shown?
The wording in this problem is complicated, but you only need to notice a few things. You can tell in the picture that this triangle has angles of , and (This assumes that the flagpole is perpendicular to the ground, but that is a safe assumption). The height of the flagpole is the longer leg in the triangle, so use the special right triangle ratios to find the length of the hypotenuse.
The longer leg is the product of the shorter leg and . The length of the shorter leg is given as , so the height of the flagpole is .
The length of the hypotenuse is the hypotenuse of a triangle. It will always be twice the length of the shorter leg, so it will equal , or .
Example 7
Antonio built a square patio in his backyard.
He wants to make a water pipe for flowers that goes from one corner to another, diagonally. How long will that pipe be?
The first step in a word problem of this nature is to add important information to the drawing. Because the problem asks you to find the length from one corner to another, you should draw that segment in.
Once you draw the diagonal path, you can see how triangles help answer this question. Because both legs of the triangle have the same measurement , this is an isosceles right triangle. The angles in an isosceles right triangle are , and .
In an isosceles right triangle, the hypotenuse is always equal to the product of the length of one leg and . So, the length of Antonio’s water pipe will be the product of and , or . This value is approximately equal to .
Lesson Summary
In this lesson, we explored how to work with different radicals both in theory and in practical situations. Specifically, we have learned:
How to identify and use the ratios involved with right isosceles triangles.
How to identify and use the ratios involved with triangles.
How to identify and use ratios involved with equilateral triangles.
How to employ right triangle ratios when solving real-world problems.
These skills will help you solve many different types of problems. Always be on the lookout for new and interesting ways to find relationships between sides and angles in triangles.
Review Questions
Mildred had a piece of scrap wood cut into an equilateral triangle. She wants to cut it into two smaller congruent triangles. What will be the angle measurement of the triangles that result?
Roberto has a square pizza. He wants to cut two congruent triangles out of the pizza without leaving any leftovers. What will be the angle measurements of the triangles that result?
What is the length of the hypotenuse in the triangle below?
What is the length of the hypotenuse in the triangle below?
What is the length of the longer leg in the triangle below?
What is the length of one of the legs in the triangle below?
What is the length of the shorter leg in the triangle below?
A square window has a diagonal of . What is the length of one of its sides?
A square block of foam is cut into two congruent wedges. If a side of the original block was , how long is the diagonal cut?
They wants to find the area of an equilateral triangle but only knows that the length of one side is . What is the height of Thuy’s triangle? What is the area of the triangle?
Review Answers
, and
, and
or approx.
or approx.
or approx.
or approx. . The area is
Tangent Ratio
Learning Objectives
Identify the different parts of right triangles.
Identify and use the tangent ratio in a right triangle.
Identify complementary angles in right triangles.
Understand tangent ratios in special right triangles.
Introduction
Now that you are familiar with right triangles, the ratios that relate the sides, as well as other important applications, it is time to learn about trigonometric ratios. Trigonometric ratios show the relationship between the sides of a triangle and the angles inside of it. This lesson focuses on the tangent ratio.
Parts of a Triangle
In trigonometry, there are a number of different labels attributed to different sides of a right triangle. They are usually in reference to a specific angle. The hypotenuse of a triangle is always the same, but the terms adjacent and opposite depend on which angle you are referencing. A side adjacent to an angle is the leg of the triangle that helps form the angle. A side opposite to an angle is the leg of the triangle that does not help form the angle.
In the triangle shown above, segment is adjacent to , and segment is opposite to . Similarly, is adjacent to , and is opposite . The hypotenuse is always .
Example 1
Examine the triangle in the
diagram below.
Identify which leg is adjacent to , opposite to , and the hypotenuse.
The first part of the question asks you to identify the leg adjacent to . Since an adjacent leg is the one that helps to form the angle and is not the hypotenuse, it must be . The next part of the question asks you to identify the leg opposite . Since an opposite leg is the leg that does not help to form the angle, it must be . The hypotenuse is always opposite the right angle, so in this triangle the hypotenuse is segment .
The Tangent Ratio
The first ratio to examine when studying right triangles is the tangent. The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side. The hypotenuse is not involved in the tangent at all. Be sure when you find a tangent that you find the opposite and adjacent sides relative to the angle in question.
For an acute angle measuring , we define .
CK-12 Geometry Page 31