7.3: Pythagoras
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Pythagoras and the Pythagorean Theorem
Pythagoras was a Greek philosopher who lived around 500 BC. He is credited as being a philosopher and mathematician. Much of what we know of Pythagoras is from writings that were copied down hundreds of years after his death, so the validity of what we do know is questionable. He is credited with Pythagoras’ theorem when actually it has been proven that Babylonians and Indians were using variations of it for hundreds of years before he even came along.
The Pythagorean theorem, also known as Pythagoras’ theorem, is a relation in Euclidean geometry among the three sides of a right triangle. ‘It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
In a right triangle with legs \(a\) and \(b\) and hypotenuse \(c\),
\[a^2+b^2=c^2 \nonumber \]
When we look at the formula, there is one important thing to remember: \(C\) is always the longest side. \(A\) and \(B\) can be swapped around, but when using this formula, \(C\) is always the longest side (which is also the side opposite the 90-degree angle).
Figure 15. Longest side triangle
Basically, the Pythagoras’ theorem says that you can figure out any side of a right triangle as long as you have the other two sides. And, if you know the lengths of all three sides of a triangle, you can use the Pythagorean theorem to verify whether the triangle is a right triangle or not. The ancient Egyptians used this method for surveying when they needed to redraw boundaries after the yearly flooding of the Nile washed away their previous markings.[1]
Video! This video walks through how to apply Pythagoras’ theorem on a right triangle.
Use the Pythagorean theorem to determine whether either of the following triangles is a right triangle.
8. 
9. 
- Answer
-
8. right triangle, because \(5^2+12^2=13^2\)
9. not a right triangle, because \(8^2+17^2\neq19^2\)
Before we continue, we need to briefly discuss square roots. Calculating a square root is the opposite of squaring a number. For example, \(\sqrt{49}=7\) because \(7^2=49\). If the number under the square root symbol is not a perfect square like \(49\), then the square root will be an irrational decimal that we will round off as necessary.
Use a calculator to find the value of each square root. Round to the hundredths place.
10. \(\sqrt{50}\)
11. \(\sqrt{296}\)
12. \(\sqrt{943}\)
- Answer
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10. \(7.07\)
11. \(17.20\)
12. \(30.71\)
We most often use the Pythagorean theorem to calculate the length of a missing side of a right triangle. Here are three different versions of the Pythagorean theorem arranged to find a missing side, so you don’t have to use algebra with \(a^2+b^2=c^2\).
The Pythagorean Theorem, three other versions
\(c=\sqrt{a^2+b^2}\)
\(b=\sqrt{c^2-a^2}\)
\(a=\sqrt{c^2-b^2}\)
Find the length of the missing side for each of these right triangles. Round to the nearest tenth, if necessary.
13. 
14. 
15. 
16. 
- Answer
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13. \(10\text{ ft}\)
14. \(15\text{ ft}\)
15. \(12.3\text{ cm}\)
16. \(1.8\text{ cm}\)
- The surveyors were called "rope-stretchers" because they used a loop of rope \(12\) units long with \(12\) equally-spaced knots. Three rope-stretchers each held a knot, forming a triangle with lengths \(3\), \(4\), and \(5\) units. When the rope was stretched tight, they knew that the angle between the \(3\)-unit and \(4\)-unit sides was a right angle because \(3^2+4^2=5^2\). From Discovering Geometry: an Inductive Approach by Michael Serra, Key Curriculum Press, 1997. ↵