Section5.3The Pythagorean Theorem and the Golden Triangle
In this section, we will see how similar right triangles can be used to derive yet another proof of the Pythagorean Theorem. We also will learn about self-similarity and the Golden Triangle. First, we review the key concepts of dilations and similarity already explored in this chapter.
Subsection5.3.1A Review of Similarity
Exploration5.3.1.
(a)
Figure 5.3.1 gives a triangle \(\Delta ABC\) and its image \(\Delta A'B'C'\text{.}\) Use either the GeoGebra applet or the coordinate system to complete the following.
Determine the center \(O\) of the dilation that takes \(\Delta ABC\) to \(\Delta A'B'C'\text{.}\)
Hint.
What lines must \(O\) lie on?
(ii)
Determine \(\frac{OA'}{OA}\text{,}\)\(\frac{OB'}{OB}\text{,}\) and \(\frac{OC'}{OC}\text{.}\) What do these ratios tell you about the dilation?
Hint.
Besides the center, what else defines a dilation?
(iii)
Is it true that \(\frac{OA'}{AA'}=\frac{OB'}{BB'}=\frac{OC'}{CC'}\) ? Are these equal to the scale factor?
(iv)
Is it true that \(\frac{A'B'}{AB}=\frac{B'C'}{BC}=\frac{A'C'}{AC}\) ? Are these equal to the scale factor?
(v)
Which pairs of angles must be congruent?
Hint.
There are three pairs in the original sketch of the two triangles. If you want an added challenge you can find more pairs using your center as a side point.
Measure angles \(\angle ABC\text{,}\)\(\angle BCA\text{,}\) and \(\angle CAB\text{.}\)
(ii)
Draw a line segment from point \(C\) meeting side \(\overline{AB}\) perpendicularly at a point \(D\text{.}\) How many triangles are now in the sketch? Name them.
(iii)
To what other angle(s) is \(\angle CDA\) congruent?
(iv)
Explain how you know triangle \(\Delta ACD\) is similar to \(\Delta ABC\text{.}\) What is the constant of proportionality?
(v)
What other triangle is similar to triangle \(\Delta ABC\text{?}\) Be sure to write the letters in the order of corresponding vertices.
(b)A Similar Triangle Proof of the Pythagorean Theorem.
We continue to use Figure 5.3.3 and the constructions from above in this task.
(i)
Explain why \(\frac{AC}{AB}=\frac{AD}{AC}\) and \(\frac{BC}{AB}=\frac{BD}{BC}\text{.}\)
(ii)
Since \(\frac{AC}{AB}=\frac{AD}{AC}\text{,}\) we see \(AC\cdot AC=AB\cdot AD\text{.}\) Why?
(iii)
Use the equation\(\frac{BC}{AB}=\frac{BD}{BC}\text{,}\) to complete the equation \(BC\cdot BC=\text{.}\)
(iv)
So that the Pythagorean Theorem will be more obvious, we will assign each length a single letter; namely, \(a=BC\text{,}\)\(b=AC\text{,}\) and \(c=AB\text{.}\) What is \(AD+DB\) using these lower-case letter(s)?
(v)
Equating \(a^2+b^2\) with \(BC\cdot BC+AC\cdot AC\text{,}\) substitute the righthand expressions from the equations in the first two parts of this task. Then add the expressions. Do you get \(c^2\text{?}\) Show all work.
Subsection5.3.3Self-Similar Triangles
The right triangle in Exploration 5.3.2 has the property that the altitude drawn from the right angle to the hypotenuse splits the triangle into two smaller triangles. Both of these triangles are similar to the original triangle. This property is true for all right triangles since the measure of the two acute angles did not matter. In particular, each new triangle has a right angle and one of the original angles. Since the sum of the angles in any triangle is 180 degrees, the newly created angle has the same measure as the remaining angle of the original triangle.
Will this happen with other triangles? We explore this question in the next activity.
Exploration5.3.3.A Special Triangle.
In Figure 5.3.4, an isosceles triangle with two 72-degree angles is provided. Use the GeoGebra applet to answer the following:
Figure5.3.4.A Golden Triangle
(a)
What is the measure of the third angle \(\angle ACB\text{?}\)
(b)
Compute \(\frac{AC}{AB}\text{,}\) the ratio of the long side to the short side. As you record this number, write down at least four digits after the decimal point. Adjust your calculator settings to show more digits if fewer than five digits are visible.
may be used to perform these calculations. If you enter your computation, the application will give you a new calculation line so you can compare results.
(c)
Use the angle bisector tool to bisect angle \(\angle CAB\text{.}\) Let \(D\) be the point where this bisector intersects side \(BC\) and create line segment \(\overline{AD}\text{.}\) Use the show/hide tool to hide bisector \(\overrightarrow{AD}\text{.}\) Segment \(\overline{AD}\) should still be visible.
(i)
Bisecting \(\angle CAB\) splits \(\Delta ABC\) into two triangles. Use GeoGebra’s Polygon tool to highlight the subtriangle that is similar to \(\Delta ABC\text{.}\)
(ii)
List the three pairs of corresponding sides.
(iii)
What is the constant of proportionality, \(k\) for this similarity? As you record this number, write down at least four digits after the decimal point.
(iv)
Note that the constant of proportionality depends on whether you chose to divide the shorter length by the longer or to divide the longer length by the shorter. Perform whichever computation have not already done and record at least four digits after the decimal point.
(v)
What do you notice about these two numbers?
Hint.
There is some slight unavoidable error in the numbers due to GeoGebra measurements and estimation. Five significant digits seems to work here.
(d)
Now use the angle bisector tool to bisect angle \(\angle ABD\) in Figure 5.3.4. Let \(E\) be the point where this angle bisector intersects segment \(\overline{AD}\text{.}\) Draw segment \(\overline{BE}\) and hide ray \(\overrightarrow{BE}\text{.}\)
(i)
Identify another triangle in the sketch which is similar to \(\Delta ABE\text{.}\)
(ii)
Identify two triangles in the sketch that are similar to \(\Delta DEB\text{.}\)
(e)Reflective Questions and Extensions.
Based on your work in this exploration, discuss the following:
(i)
If you use the arrow tool to move vertex \(A\) or \(B\text{,}\) the lengths of the sides of triangles will change, but the angle measures will not. Do the constants of proportionality change?
(ii)
Suppose you continued this process, bisecting \(\angle BDE\) to create triangle \(\Delta DEF\text{,}\) then bisecting \(\angle DEF\) to create \(\Delta EFG\text{,}\) and so forth. What type of design would you get? What properties would it have?
(iii)
The area tool is under the angle menu. What is the ratio of the areas of similar triangles \(\Delta ABC\) and \(\Delta BDA\text{?}\) How is this number related to the ratio of the corresponding sides?
Hint.
Use the ‘Polygon’ tool to create \(\Delta BDA\) before using the GeoGebra ‘Area’ tool.
Use a calculator to estimate the number \(\frac{1+\sqrt{5}}{2}\text{.}\) What do you notice?
The isosceles triangle \(\Delta ABC\) explored in Exploration 5.3.3 is known as a Golden Triangle. When we create a smaller, but similar, triangle by bisecting a base angle, the constant of proportionality equals the ratio of the long side to the short side (base) of the original triangle. We can repeat this process of bisecting the base angle of our new triangle to form even smaller similar triangles indefinitely. At every stage of the process, the ratio of the long side to the base will remain \(\frac{1+\sqrt{5}}{2}\approx 1.618033989\) and equal to the constant of proportionality for the pair of similar triangles. This number is known as the Golden Ratio and is an irrational number 3
An irrational number cannot be written as the ratio of integers. The decimal expansion of an irrational number is infinite and does not become an infinitely repeating block of digits.
The polygon \(ABCDE\) in Figure 5.3.5 is a regular pentagon.
Figure5.3.5.A Triangle in a Regular Pentagon
(a)
In Section 2.4, we learned that each vertex angle of a regular \(n\)-gon measures \(\frac{180(n-2)}{n}\) degrees. Assume that \(ABCDE\) is a regular pentagon and determine the measure of \(\angle EAB\text{.}\)
(b)
Draw segment \(\overline{AD}\text{.}\) What type of triangle is \(\Delta EAD\text{?}\)
(c)
Compute \(m\angle{EAD}\text{.}\) How did you determine this?
(d)
Determine \(m\angle{DAB}\) showing all work.
(e)
Construct \(\overline{BD}\text{.}\) What is the measure of \(\angle CDE\text{?}\) How do you know?
(f)
Describe triangle \(\Delta DAB\) as specifically as possible. Why does this exploration belong in this section?
(g)
Now extend lines \(\overleftrightarrow{ED}\) and \(\overleftrightarrow{BC}\) so that they meet at a point \(F\text{.}\) Similarly, let \(G\) be the point where \(\overleftrightarrow{DC}\) meets \(\overleftrightarrow{AB}\text{,}\) let \(H\) be the point where \(\overleftrightarrow{CB}\) and \(\overleftrightarrow{EA}\) intersect, let \(I\) be the intersection of \(\overleftrightarrow{BA}\) and \(\overleftrightarrow{DE}\text{,}\) and let \(J\) be the intersection of \(\overleftrightarrow{AE}\) and \(\overleftrightarrow{CD}\text{.}\)
(i)
Describe the shape of the decagon \(DFCGBHAIEJ\text{.}\)
(ii)
What type of triangle is \(\Delta DFC\text{?}\) How is it related to \(\Delta ADB\text{?}\)
Exercises5.3.5Exercises
Skills and Recall
1.
In Figure 5.3.6, \(D\text{,}\)\(E\text{,}\) and \(F\) are the midpoints of \(\overline{AB}\text{,}\)\(\overline{BC}\text{,}\) and \(\overline{AC}\text{,}\) respectively. Also, \(AB=8\text{,}\)\(BC=14\text{,}\) and \(AC=10\text{.}\) Angles \(\angle ABC\text{,}\)\(\angle ADF\text{,}\) and \(\angle FEC\) have measure 42 degrees; whereas, \(m\angle ACB=30\deg\) and \(m\angle DEF=108\deg\text{.}\) Answer the following:
Figure5.3.6.
(a)
Determine \(m\angle DEB\) and \(m\angle BAC\text{.}\)
(b)
Determine \(m\angle EFC\text{,}\)\(m\angle BDE\text{,}\) and \(m\angle BAC\text{.}\)
(c)
Determine \(m\angle FDE\) and \(m\angle AFD\text{.}\)
(d)
Determine \(m\angle EFD\text{.}\)
(e)
Identify four triangles that are similar to \(\Delta ABC\text{.}\) Be sure to list the vertices in order.
(f)
Determine the length of \(\overline{AF}\text{.}\) Support your answer.
(g)
Determine the length of \(\overline{DF}\text{.}\) Support your answer.
(h)
There is a dilation taking \(\Delta ABC\) to \(\Delta FEC\text{.}\) What is the center and scale factor of this dilation?
(i)
There is an isometry taking \(\Delta DEB\) to \(\Delta FCE\text{.}\) What type of isometry? Identify its defining object(s).
