4.4: Similar Triangles and Pythagorean Theorem
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)This is another excerpt from Raphael’s The School of Athens. The man writing in the book represents Pythagoras, the namesake of one of the most widely used formulas in geometry, engineering, architecture, and many other fields, the Pythagorean Theorem. However, there is evidence that the theorem was known as early as \(1900–1100\) BC by the Babylonians. The Pythagorean Theorem is a formula used for finding the lengths of the sides of right triangles.
Born in Greece, Pythagoras lived from \(569–500\) BC. He initiated a cult-like group called the Pythagoreans, which was a secret society composed of mathematicians, philosophers, and musicians. Pythagoras believed that everything in the world could be explained through numbers. Besides the Pythagorean Theorem, Pythagoras and his followers are credited with the discovery of irrational numbers, the musical scale, the relationship between music and mathematics, and many other concepts that left an immeasurable influence on future mathematicians and scientists.
The focus of this section is on right triangles. We will look at how the Pythagorean Theorem is used to find the unknown sides of a right triangle, and we will also study the special triangles, those with set ratios between the lengths of sides. By ratios we mean the relationship of one side to another side. When you think about ratios, you should think about fractions. A fraction is a ratio, the ratio of the numerator to the denominator. Finally, we will preview trigonometry. We will learn about the basic trigonometric functions, sine, cosine and tangent, and how they are used to find not only unknown sides but unknown angles, as well, with little information.
Pythagorean Theorem
The Pythagorean Theorem is used to find unknown sides of right triangles. The theorem states that the sum of the squares of the two legs of a right triangle equals the square of the hypotenuse (the longest side of the right triangle).
The Pythagorean Theorem states
\[a^2+b^2=c^2 \nonumber \]
where \(a\) and \(b\) are two sides (legs) of a right triangle and \(c\) is the hypotenuse, as shown in Figure \(\PageIndex{2}\). Note that \(c\) is longer than legs \(a\) and \(b\).
So we do have a formula for one of the sides if the other two are known. The following three formulas can be obtained from \(a^2+b^2=c^2 \).
\[\begin{aligned} c&= \sqrt{a^2+b^2}\\
a&= \sqrt{c^2-b^2}\\
b&= \sqrt{c^2-a^2}\\
\end{aligned} \nonumber \]
Find the length of the missing side of the triangle in Figure \(\PageIndex{3}\).
- Answer
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So we do have a formula for one of the sides if the other two are known\[\begin{aligned}
b&= \sqrt{c^2-a^2}\\
&= \sqrt{14^2-6^2}\\
&= \sqrt{196-36}\\
&= \sqrt{160}\\
&= \sqrt{16\times10}\\
&= 4\sqrt{10}~~~~~~ {\text{~~Exact answer}}\\
&=12.65~~~~~~ {\text{~~Approximate answer}}\\
\end{aligned} \nonumber \].
You live on the corner of First Street and Maple Avenue, and work at Star Enterprises on Tenth Street and Elm Drive (Figure ="lt-icon-d\(\PageIndex{4}\) ). You want to calculate how far you walk to work every day and how it compares to the actual distance. Each block measures \(200\) ft by \(200\) ft.
- Answer
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You travel \(7\) blocks south and \(9\) blocks west. If each block measures \(200\) ft by \(200\) ft, so everyday walk will be
\[9(200)+7(200)=1,800 \mathrm{~ft}+1,400 \mathrm{~ft}=3,200 \mathrm{~ft}\nonumber\]
The actual distance can be found using the Pythagorean theorem
\[\begin{aligned}
d & =\sqrt{(1,800)^2+(1,400)^2} \\
& =\sqrt{3,240,000+1,960,000} \\
& =\sqrt{5,200,000} \\
& =2280.4 \mathrm{~ft}
\end{aligned} \nonumber \]
TV screens are measured on the diagonal. If we have a TV-cabinet that is \(76\) inches long and \(64\) inches high, how large a TV could we put in the space (leave \(2\)-inches on all sides for the edging of the TV)? Round your answer to the nearest hundredth.
- Answer
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If we leave a space of \(2\) inches on all sides, the length of the TV must be \(76-4=72\) inches and the width must be \(64-4=60\) inches. So to find the size of the TV, we need to find the diagonal of the TV, which is the hypotenuse of a right triangle with legs \(72\) inches and \(60\) inches.
\[\begin{aligned}
c&= \sqrt{a^2+b^2}\\
&= \sqrt{72^2+60^2}\\
&= \sqrt{5184+3600}\\
&= \sqrt{8784}\\
&=93.72
\end{aligned} \nonumber \]So \(93.72\) inch TV can be placed in the cabinet.
The Pythagorean Theorem is so widely used that most people assume that Pythagoras (\(570–490\) BC) discovered it. The philosopher and mathematician uncovered evidence of the right triangle concepts in the teachings of the Babylonians dating around 1900 BC. However, it was Pythagoras who found countless applications of the theorem leading to advances in geometry, architecture, astronomy, and engineering.
Among his accolades, Pythagoras founded a school for the study of mathematics and music. Students were called the Pythagoreans, and the school’s teachings could be classified as a religious indoctrination just as much as an academic experience. Pythagoras believed that spirituality and science coexist, that the intellectual mind is superior to the senses, and that intuition should be honored over observation.
Pythagoras was convinced that the universe could be defined by numbers, and that the natural world was based on mathematics. His primary belief was All is Number. He even attributed certain qualities to certain numbers, such as the number 8 represented justice and the number 7 represented wisdom. There was a quasi-mythology that surrounded Pythagoras. His followers thought that he was more of a spiritual being, a sort of mystic that was all-knowing and could travel through time and space. Some believed that Pythagoras had mystical powers, although these beliefs were never substantiated.
Pythagoras and his followers contributed more ideas to the field of mathematics, music, and astronomy besides the Pythagorean Theorem. The Pythagoreans are credited with the discovery of irrational numbers and of proving that the morning star was the planet Venus and not a star at all. They are also credited with the discovery of the musical scale and that different strings made different sounds based on their length. Some other concepts attributed to the Pythagoreans include the properties relating to triangles other than the right triangle, one of which is that the sum of the interior angles of a triangle equals These geometric principles, proposed by the Pythagoreans, were proven \(200\) years later by Euclid.
The Pythagorean Theorem is perhaps one of the most useful formulas you will learn in mathematics because there are so many applications of it in real world settings. Architects and engineers frequently use this formula when designing ramps, bridges, and buildings.
Similar Triangles
Two triangles are said to be similar if they have equal sets of angles. In Figure \(\PageIndex{5}\), \(\triangle ABC\) is similar to \(\triangle DEF \). The angles that are equal are called corresponding angles. In figure \(\PageIndex{5}\),
\[\begin{aligned}
\angle A& \text{ corresponds to } \angle D\\
\angle B & \text{ corresponds to } \angle E\\
\angle C & \text{ corresponds to } \angle F\\
\end{aligned} \nonumber \]
The sides joining corresponding vertices are called corresponding sides. In figure \(\PageIndex{5\)
\[\begin{aligned}
AB & \text{ corresponds to } DE\\
BC & \text{ corresponds to } EF\\
AC & \text{ corresponds to } DF\\
\end{aligned} \nonumber \]
The symbol for similar is \(\sim\). The similarity statement \(\triangle ABC \sim \triangle DEF\) will always be written so that corresponding vertices appear in the same order.
For the triangles in Figure \(\PageIndex{5}\), we could also write
\[\begin{aligned}
\triangle BAC &\sim \triangle BDF\\
\triangle ACB &\sim \triangle DFE
\end{aligned} \nonumber \]
But never
\[\begin{aligned}
\triangle ABC &\sim \triangle EDF\\
\triangle ACB &\sim \triangle DEF\\
\end{aligned} \nonumber \]
We can determine which sides correspond based on the similarity statement. For example, if \(\triangle ABC \sim \triangle DEF\), then side \(AB\) corresponds to side \(DE\) because both are the first two letters of each triangle. \(BC\) corresponds to \(EF\) because both are the last two letters of a triangle, and \(AC\) corresponds to \(DF\) because both consist of the first and last letters.
If two triangles have all three corresponding angles equal, then the triangles are similar.
\(\triangle ABC \sim \triangle DEF\) if
\[\angle A = \angle D ~\text{and}~\angle B =\angle D= 60^{\circ} \nonumber\]
We see that the above triangles are similar because
\[\angle B = 180^{\circ} - (45^{\circ} + 30^{\circ}) = 180^{\circ} - 75^{\circ} = 105^{\circ}=\angle E \nonumber\]
Two triangles are similar if either their corresponding sides are proportional.
\(\triangle UVW \sim \triangle XYZ\) if
\( \dfrac{\overline{UV}}{\overline{XY}} = \dfrac{\overline{VW}}{\overline{YZ}} = \dfrac{\overline{WU}}{\overline{ZX}} \)
We can see that \(\triangle UVW \sim \triangle XYZ\). Substitute the side lengths into the ratios, and determine if the ratios of the corresponding sides are equivalent. They are, so the triangles are similar.
\( \dfrac{6}{4} = \dfrac{12}{8} = \dfrac{9}{6} \)
\(1.5 = 1.5 = 1.5\)
Two triangles are similar if the ratio of two pairs of corresponding sides is equal and the angles between those sides (the included angles) are congruent (equal).
Figure \(\PageIndex{8}\): SAS Similarity
\(\triangle ABC \sim \triangle DEF\) if
\[\dfrac{AC}{DF} = \dfrac{AB}{DE}~\text{and}~\angle A =\angle D= 60^{\circ} \nonumber\]
Finding Missing Measurements in Similar Triangles
You can find the missing measurements in a triangle if you know some measurements of a similar triangle. Let’s look at an example.
Find the unknown lengths for the similar triangles below.
- Answer
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In similar triangles, the ratios of corresponding sides are proportional. Set up a proportion of two ratios, one that includes the missing side.
\(\dfrac{3}{4.2} = \dfrac{12}{y}=\dfrac{11.5}{x}\)
Choose the first and second ratios to solve for \(y\)
\(\dfrac{3}{4.2} = \dfrac{12}{y}\)
Solve for \(y\)using cross multiplication.
\[\begin{aligned}
3 \times y&= 4.2\times 12\\
3y &= 50.4\\
y &= \frac{50.4}{3}\\
&y=16.8\\
\end{aligned} \nonumber \]Choose the first and third ratios to solve for \(x\)
\(\dfrac{3}{4.2} = \dfrac{11.5}{x}\)
Solve for \(x\)using cross multiplication.
\[\begin{aligned}
3 \cdot x&= 11.5\cdot 4.2\\
3x&= 48.3\\
x&=\frac{48.3}{3}\\
x &= 16.1\\
\end{aligned} \nonumber \]
This process is fairly straightforward—but be careful that your ratios represent corresponding sides, recalling that corresponding sides are opposite corresponding angles.
Solving Application Problems Involving Similar Triangles
Applying knowledge of triangles, similarity, and congruence can be very useful for solving problems in real life. Just as you can solve for missing lengths of a triangle drawn on a page, you can use triangles to find unknown distances between locations or objects.
Let’s consider the example of two trees and their shadows. Suppose the sun is shining down on two trees, one that is 6 feet tall and the other whose height is unknown. By measuring the length of each shadow on the ground, you can use triangle similarity to find the unknown height of the second tree.
First, let’s figure out where the triangles are in this situation. The trees themselves create one pair of corresponding sides. The shadows cast on the ground are another pair of corresponding sides. The third side of these imaginary similar triangles runs from the top of each tree to the tip of its shadow on the ground. This is the hypotenuse of the triangle.
If you know that the trees and their shadows form similar triangles, you can set up a proportion to find the height of the tree.
When the sun is at a certain angle in the sky, a \(6\)-foot tree will cast a \(4\)-foot shadow. How tall is a tree that casts an \(8\)-foot shadow?
- Answer
-
The angle measurements are the same, so the triangles are similar triangles. Since they are similar triangles, you can use proportions to find the size of the missing side. Set up a proportion comparing the heights of the trees and the lengths of their shadows.
\(\dfrac{\text{Height of Tree 1}}{\text{Height of Tree 2}} = \dfrac{\text{Length of Shadow of Tree 1}}{\text{Length of Shadow of Tree 2}}\)
Substitute in the known lengths. Call the missing tree height h.
\(\dfrac{6}{h} = \dfrac{4}{8}\)
Solve for h using cross-multiplication.
\(6 \cdot 8 = 4h\)
\(48 = 4h\)
\(12 = h\)
The tree is \(12\) feet tall.
If a \(30\)-meter flagpole casts a shadow that is \(50\) meters long, how long is the shadow cast by a tree that is \(36\) meters high? Ans 60 meters
- Answer
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\[\dfrac{\text{flagpole height }}{\text{tree height}} = \dfrac{\text{flagpole shadow}}{\text{tree shadow}}\nonumber\]
Substitute in the known lengths. Call the missing tree shadow, l.
\(\dfrac{30}{36} = \dfrac{50}{l}\)
Solve for h using cross-multiplication.
\(50 \cdot 36 = 30l\)
\(180 = 30l\)
\(60= l\)
So \(36\) meters high tree cast \(60\) meter shadow.