If you are given the base and the sides of an isosceles triangle, you can use the Pythagorean Theorem to calculate the height. Recall that the height (altitude) of a triangle is the length of a segment from one angle in the triangle perpendicular to the opposite side. In this case we focus on the altitude of isosceles triangles going from the vertex angle to the base.
Example 4
What is the height of the triangle below?
To find the area of this isosceles triangle, you will need to know the height in addition to the base. Draw in the height by connecting the vertex of the triangle with the base at a right angle.
Since the triangle is isosceles, the altitude will bisect the base. That means that it will divide it into two congruent, or equal parts. So, you can identify the length of one half of the base as .
If you look at the smaller triangle now inscribed in the original shape, you’ll notice that it is a right triangle with one leg and hypotenuse So, this is a triangle. If the leg is and the hypotenuse is , the missing leg must be . So, the height of the isosceles triangle is .
Use this information along with the original measurement of the base to find the area of the entire isosceles triangle.
The area of the entire isosceles triangle is .
The Distance Formula
You have already learned that you can use the Pythagorean Theorem to understand different types of right triangles, find missing lengths, and identify Pythagorean triples. You can also apply the Pythagorean Theorem to a coordinate grid and learn how to use it to find distances between points.
Example 5
Look at the points on the grid below.
Find the length of the segment connecting and .
The question asks you to identify the length of the segment. Because the segment is not parallel to either axis, it is difficult to measure given the coordinate grid. However, it is possible to think of this segment as the hypotenuse of a right triangle. Draw a vertical line at and a horizontal line at and find the point of intersection. This point represents the third vertex in the right triangle.
You can easily count the lengths of the legs of this triangle on the grid. The vertical leg extends from to , so it is long. The horizontal leg extends from to , so it is long. Use the Pythagorean Theorem with these values for the lengths of each leg to find the length of the hypotenuse.
The segment connecting and is long.
Mathematicians have simplified this process and created a formula that uses these steps to find the distance between any two points in the coordinate plane. If you use the distance formula, you don’t have to draw the extra lines.
Distance Formula: Give points and , the length of the segment connecting those two points is
Example 6
Use the distance formula to find the distance between the points and on a coordinate grid.
You already know from example 1 that the distance will be , but you can practice using the distance formula to make sure it works. In this formula, substitute for for for , and for because and are the two points in question.
Now you see that no matter which method you use to solve this problem, the distance between and on a coordinate grid is .
Lesson Summary
In this lesson, we explored how to work with different radical expressions both in theory and in practical situations. Specifically, we have learned:
How to identify and employ the Pythagorean Theorem when working with right triangles.
How to identify common Pythagorean triples.
How to use the Pythagorean Theorem to find the area of isosceles triangles.
How to use the Pythagorean Theorem to derive the distance formula on a coordinate grid.
These skills will help you solve many different types of problems. Always be on the lookout for new and interesting ways to apply the Pythagorean Theorem to mathematical situations.
Points to Consider
Now that you have learned the Pythagorean Theorem, there are countless ways to apply it. Could you use the Pythagorean Theorem to prove that a triangle contained a right angle if you did not have an accurate diagram?
Review Questions
What is the distance between and ?
Do the numbers , , and make a Pythagorean triple?
What is the length of in the triangle below?
Do the numbers , and make a Pythagorean triple?
What is the distance between and ?
What is the length of in the triangle below?
What is the distance between and ?
What is the area of below?
What is the area of the triangle below?
What is the area of the triangle below?
An alternative proof of the Pythagorean Theorem uses the area of a square. The diagram below shows a square with side lengths , and an inner square with side lengths . Use the diagram below to prove Hint: Find the area of the inner square in two ways: once directly, and once by finding the area of the larger square and subtracting the area of each triangle.
Review Answers
yes
no
Proof. The plan is, we will find the area of the green square in two ways. Since those two areas must be equal, we can set those areas equal to each other. For the inner square (in green), we can directly compute the area: .
Now, the area of the large, outer square is . Don’t forget to multiply this binomial carefully!
The area of each small right triangle (in yellow) is
.
Since there are four of those right triangles, we have the combined area
Finally, subtract the area of the four yellow triangles from the area of the larger square, and we are left with
Putting together the two different ways for finding the area of the inner square, we have
Converse of the Pythagorean Theorem
Learning Objectives
Understand the converse of the Pythagorean Theorem.
Identify acute triangles from side measures.
Identify obtuse triangles from side measures.
Classify triangles in a number of different ways.
Converse of the Pythagorean Theorem
In the last lesson, you learned about the Pythagorean Theorem and how it can be used. As you recall, it states that the sum of the squares of the legs of any right triangle will equal the square of the hypotenuse. If the lengths of the legs are labeled and , and the hypotenuse is , then we get the familiar equation:
The Converse of the Pythagorean Theorem is also true. That is, if the lengths of three sides of a triangle make the equation true, then they represent the sides of a right triangle.
With this converse, you can use the Pythagorean Theorem to prove that a triangle is a right triangle, even if you do not know any of the triangle’s angle measurements.
Example 1
Does the triangle below contain a right angle?
This triangle does not have any right angle marks or measured angles, so you cannot assume you know whether the triangle is acute, right, or obtuse just by looking at it. Take a moment to analyze the side lengths and see how they are related. Two of the sides and are relatively close in length. The third side is about half the length of the two longer sides.
To see if the triangle might be right, try substituting the side lengths into the Pythagorean Theorem to see if they makes the equation true. The hypotenuse is always the longest side, so should be substituted for . The other two values can represent and and the order is not important.
Since both sides of the equation are equal, these values satisfy the Pythagorean Theorem. Therefore, the triangle described in the problem is a right triangle.
In summary, example 1 shows how you can use the converse of the Pythagorean Theorem. The Pythagorean Theorem states that in a right triangle with legs and , and hypotenuse , . The converse of the Pythagorean Theorem states that if , then the triangle is a right triangle.
Identifying Acute Triangles
Using the converse of the
Pythagorean Theorem, you can identify whether triangles contain a right angle or not. However, if a triangle does not contain a right angle, you can still learn more about the triangle itself by using the formula from Pythagorean Theorem. If the sum of the squares of the two shorter sides of a triangle is greater than the square of the longest side, the triangle is acute (all angles are less than ). In symbols, if then the triangle is acute.
Identifying the "shorter" and "longest" sides may seem ambiguous if sides have the same length, but in this case any ordering of equal length sides leads to the same result. For example, an equilateral triangle always satisfies and so is acute.
Example 2
Is the triangle below acute or right?
The two shorter sides of the triangle are and . The longest side of the triangle is . First find the sum of the squares of the two shorter legs.
The sum of the squares of the two shorter legs is Compare this to the square of the longest side,
The square of the longest side is Since , this triangle is not a right triangle. Compare the two values to identify which is greater.
The sum of the square of the shorter sides is greater than the square of the longest side. Therefore, this is an acute triangle.
Identifying Obtuse Triangles
As you have probably figured out, you can prove a triangle is obtuse (has one angle larger than ) by using a similar method. Find the sum of the squares of the two shorter sides in a triangle. If this value is less than the square of the longest side, the triangle is obtuse. In symbols, if , then the triangle is obtuse. You can solve this problem in a manner almost identical to example 2 above.
Example 3
Is the triangle below acute or obtuse?
The two shorter sides of the triangle are and The longest side of the triangle is First find the sum of the squares of the two shorter legs.
The sum of the squares of the two shorter legs is Compare this to the square of the longest side,
The square of the longest side is . Since , this triangle is not a right triangle. Compare the two values to identify which is greater.
Since the sum of the square of the shorter sides is less than the square of the longest side, this is an obtuse triangle.
Triangle Classification
Now that you know the ideas presented in this lesson, you can classify any triangle as right, acute, or obtuse given the length of the three sides. Begin by ordering the sides of the triangle from smallest to largest, and substitute the three side lengths into the equation given by the Pythagorean Theorem using . Be sure to use the longest side for the hypotenuse.
If , the figure is a right triangle.
If , the figure is an acute triangle.
If , the figure is an obtuse triangle.
Example 4
Classify the triangle below as right, acute, or obtuse.
The two shorter sides of the triangle are and . The longest side of the triangle is . First find the sum of the squares of the two shorter legs.
The sum of the squares of the two shorter legs is Compare this to the square of the longest side,
The square of the longest side is Therefore, the two values are not equal, and this triangle is not a right triangle. Compare the two values, and to identify which is greater.
Since the sum of the square of the shorter sides is greater than the square of the longest side, this is an acute triangle.
Example 5
Classify the triangle below as right, acute, or obtuse.
The two shorter sides of the triangle are and . The longest side of the triangle is . First find the sum of the squares of the two shorter legs.
The sum of the squares of the two legs is . Compare this to the square of the longest side, .
The square of the longest side is Since these two values are equal, , and this is a right triangle.
Lesson Summary
In this lesson, we explored how to work with different radical expressions both in theory and in practical situations. Specifically, we have learned:
How to use the converse of the Pythagorean Theorem to prove a triangle is right.
How to identify acute triangles from side measures.
How to identify obtuse triangles from side measures.
How to classify triangles in a number of different ways.
These skills will help you solve many different types of problems. Always be on the lookout for new and interesting ways to apply the Pythagorean Theorem and its converse to mathematical situations.
Points to Consider
Use the Pythagorean Theorem to explore relationships in common right triangles. Do you find that the sides are proportionate?
Review Questions
Solve each problem.
For exercises 1-8, classify the following triangle as acute, obtuse, or right based on the given side lengths. Note, the figure is not to scale.
In the triangle below, which sides should you use for the legs (usually called sides , and ) and the hypotenuse (usually called side ), in the Pythagorean theorem? How do you know?
Review Answers
Right
Acute
Obtuse
Acute
Right
Acute
Obtuse
Obtuse
The side with length should be the hypotenuse since it is the longest side. The order of the legs does not matter
Using Similar Right Triangles
Learning Objectives
Identify similar triangles inscribed in a larger triangle.
Evaluate the geometric mean of various objects.
Identify the length of an altitude using the geometric mean of a separated hypotenuse.
Identify the length of a leg using the geometric mean of a separated hypotenuse.
Introduction
In this lesson, you will study figures inscribed, or drawn within, existing triangles. One of the most important types of lines drawn within a right triangle is called an altitude. Recall that the altitude of a triangle is the perpendicular distance from one vertex to the opposite side. By definition each leg of a right triangle is an altitude. We can find one more altitude in a right triangle by adding an auxiliary line segment that connects the vertex of the right angle with the hypotenuse, forming a new right angle.
You may recall this is the figure that we used to prove the Pythagorean Theorem. In right triangle above, the segment is an altitude. It begins at angle , which is a right angle, and it is perpendicular to the hypotenuse . In the resulting figure, we have three right triangles, and all of them are similar.
Inscribed Similar Triangles
You may recall that if two objects are similar, corresponding angles are congruent and their sides are proportional in length. In other words, similar figures are the same shape, but different sizes. To prove that two triangles are similar, it is sufficient to prove that all angle measures are congruent (note, this is NOT true for other polygons. For example, both squares and “long” rectangles have all angles, but they are not similar). Use logic, and the information presented above to complete Example 1.
Example 1
Justify the statement that .
In the figure above, the big triangle is a right triangle with right angle and and . So, if , and are similar, they will all have angles of , and .
First look at . , and . Since the sum of the three angles in a triangle always equals , the missing angle, , must measure , since . Lining up the congruent angles, we can write .
Now look at has a measure of , and . Since the sum of the three angles in a triangle always equals , the missing angle, , must measure , since Now, since the triangles have congruent corresponding angles, and are similar.
Thus, . Their angles are congruent and their sides are proportional.
Note that you must be very careful to match up corresponding angles when writing triangle similarity statements. Here we should write . This example is challenging because the triangles are overlapping.
Geometr
ic Means
When someone asks you to find the average of two numbers, you probably think of the arithmetic mean (average). Chances are good you’ve worked with arithmetic means for many years, but the concept of a geometric mean may be new. An arithmetic mean is found by dividing the sum of a set of numbers by the number of items in the set. Arithmetic means are used to calculate overall grades and many other applications. The big idea behind the arithmetic mean is to find a “measure of center” for a group of numbers.
A geometric mean applies the same principles, but relates specifically to size, length, or measure. For example, you may have two line segments as shown below. Instead of adding and dividing, you find a geometric mean by multiplying the two numbers, then finding the square root of the product.
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