The Pythagorean Theorem and the Golden Triangle

Section 5.3 The 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.

Subsection 5.3.1 A Review of Similarity

Exploration 5.3.1.

(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.

Figure 5.3.1. GeoGebra applet for Task 5.3.1.a

(i)

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?

Hint.

Recall that the scale factor is a ratio. It compares the lengths of corresponding sides of two similar shapes. Do the given ratios above compare corresponding sides?

(iv)

Is it true that \(\frac{A'B'}{AB}=\frac{B'C'}{BC}=\frac{A'C'}{AC}\) ? Are these equal to the scale factor?

Hint.

Remember that the scale factor is a ratio that compares corresponding sides of two similar shapes. Again, determine if the given ratios above compare corresponding sides.

(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.

(b)

In Figure 5.3.2, \(\Delta FGH\sim\Delta IJK\)

The triangles are similar

. Also, \(FG=4.4, FH=13, IK=7.8, JK=9.06, m\angle GFH=110.34^{\circ}\) and \(m\angle IJK=53.8^{\circ}\text{.}\) Determine the following:

Triangles FGH and IJK are similar. Angle GFH has measure 110.34, angle IJK has measure 53.8. FH has length 13, IK has length 7.8, and, JK has length 9.06. — Figure 5.3.2. Illustration for Task 5.3.1.b

(i)

\(m\angle FGH\)

Hint.

The corresponding angles of similar triangles are congruent, meaning they have the same measure. Use the order of the letters in the similarity statement \(\Delta FGH\sim\Delta IJK\) to identify the angle in the second triangle that corresponds to \(m\angle G\text{.}\)

(ii)

\(m\angle FHG\)

Hint.

To find the angle that corresponds to \(\angle H\text{,}\) once again use the similarity statement. See which angle lines up with \(\angle H\) and is congruent.

(iii)

\(m\angle JIK\)

Hint.

Use the same technique as the problems above. Use the similarity statement to determine which angle corresponds to \(\angle I\text{.}\)

(iv)

\(m\angle KIJ\)

Hint.

See previous hints that discuss similarity and congruency for angles.

(v)

The scale factor of the dilation that takes \(\Delta FGH\) to \(\Delta IJK\)

Hint.

The scale factor is the ratio of corresponding sides. Use the similarity statement, \(\Delta FGH\sim\Delta IJK\text{,}\) to identify which sides match up, then find their lengths. The ratio of their lengths represents the scale factor.

(vi)

\(GH\)

Hint.

To find the length of side \(GH\text{,}\) use the similarity statement. Find its corresponding side in \(\Delta IJK\) and apply the scale factor.

(vii)

\(IJ\)

Hint.

To find the length of side \(IJ\text{,}\) once again use the similarity statement. Find its corresponding side in \(\Delta FGH\) and apply the scale factor.

Subsection 5.3.2 Applications of Similarity

We now apply our knowledge of similarity to two topics: another proof of the Pythagorean Theorem and a triangle known as the Golden Triangle.

Exploration 5.3.2. A Proof of the Pythagorean Theorem.

Use Figure 5.3.3 for this exploration.

Figure 5.3.3. GeoGebra applet of a Right Triangle

(a) Right Triangles within a Right Triangle.

Complete the following:

(i)

Measure angles \(\angle ABC\text{,}\) \(\angle BCA\text{,}\) and \(\angle CAB\text{.}\)

Hint.

To find the measure of each of these angles, use the interactive GeoGebra tool (Figure 5.3.3). On the options bar at the top, click on the sixth button from the left, indicated by a small red angle symbol. Then proceed by clicking on the three vertices of the angle. Remember that the second point clicked should be the vertex of the angle. If the measurement given is the larger, outside angle, simply subtract it from 360 to find the interior angle.

(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.

Hint.

Think of this line segment from point \(C\) to \(\overline{AB}\) as the altitude. How does it divide the larger triangle into two smaller right triangles?

(iii)

To what other angle(s) is \(\angle CDA\) congruent?

Hint.

Look for angles that have the same orientation and are within a similar triangle.

(iv)

Explain how you know triangle \(\Delta ACD\) is similar to \(\Delta ABC\text{.}\) What is the constant of proportionality?

Hint.

To prove that these two triangles are similar, show two of their angles are congruent using the Angle-Angle (AA) rule. Recall the definition for constant of proportionality.

(v)

What other triangle is similar to triangle \(\Delta ABC?\) Be sure to write the letters in the order of corresponding vertices.

Hint.

There is another small right triangle that is formed by the perpendicular. This small right triangle is similar to \(\Delta ABC\text{.}\)

(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 \(\frac{BC}{AB}=\frac{BD}{BC}\text{.}\)

Hint.

Use the property of similar triangles which states that corresponding sides are proportional.

(ii)

Since \(\frac{AC}{AB}=\frac{AD}{AC}\text{,}\) we see \(AC\cdot AC=AB\cdot AD\text{.}\) Why?

Hint.

Think about what happens when two ratios are equal. How is cross multiplication being applied here?

(iii)

Use the equation\(\frac{BC}{AB}=\frac{BD}{BC}\text{,}\) to complete the equation \(BC\cdot BC=\text{.}\)

Hint.

Use cross multiplication to complete this equation.

(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)?

Hint.

Look at the line segment that includes both \(AD\) and \(DB\text{.}\) What is the name of that entire line segment? By which lowercase letter is it represented?

(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 \(c^2?\) Show all work.

Hint.

Recall the expressions for \(a^2\) and \(b^2\) found previously. Substitute those expressions into the equation, then add them up. Observe and take note of the result.

Subsection 5.3.3 Self-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.

Exploration 5.3.3. 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:

Figure 5.3.4. A Golden Triangle

(a)

What is the measure of the third angle \(\angle ACB?\)

Hint.

Use the Triangle Angle Sum Theorem to find \(\angle ACB\text{.}\)

(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.

Hint.

The Desmos Scientific Calculator at desmos.com

desmos.com/scientific

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 \(\Delta ABC\text{.}\)

Hint.

After bisecting \(\angle CAB\text{,}\) check to see which small triangle has identical angles to the original triangle, \(\Delta ABC\text{.}\)

(ii)

List the three pairs of corresponding sides.

Hint.

Match equal angles to pair the 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.

Hint.

Review the definition for constant of proportionality.

(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 has not already been done and record at least four digits after the decimal point.

Hint.

Compute the reciprocal of the ratio in the previous problem.

(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 \(\Delta ABE\text{.}\)

Hint.

The new bisector at angle \(\Delta ABD\) creates another isosceles triangle \(\Delta ABE\text{.}\) Which other triangle in the sketch is similar to \(\Delta ABE?\)

(ii)

Identify two triangles in the sketch that are similar to \(\Delta DEB\text{.}\)

Hint.

Look for two triangles that have identical interior angle measurements 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?

Hint.

Again, review the definition of constant of proportionality. Experiment with moving the vertices \(A\) and \(B\text{.}\) Use the ratio of corresponding sides to see if changing side lengths has any impact on the constant of proportionality.

(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?

Hint.

Notice how bisection creates a smaller triangle similar to the previous one. Imagine how the pattern would look if this process was continued.

(iii)

The area tool is under the angle menu. What is the ratio of the areas of similar triangles \(\Delta ABC\) and \(\Delta BDA?\) 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.

Also, review Section 3.5 for one relationship.

(iv)

Use a calculator to estimate the number \(\frac{1+\sqrt{5}}{2}\text{.}\) What do you notice?

Hint.

The Desmos Scientific Calculator at desmos.com

desmos.com/scientific

can be used for this estimation.

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

⁴

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.

Subsection 5.3.4 A Connection to a Regular Pentagon

We explored regular polygons in Section 2.4.

Exploration 5.3.4.

The polygon \(ABCDE\) in Figure 5.3.5 is a regular pentagon.

Regular pentagon ABCDE — Figure 5.3.5. 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 \(\angle EAB\text{.}\)

Hint.

The variable \(n\) represents the number of sides.

(b)

Draw segment \(\overline{AD}\text{.}\) What type of triangle is \(\Delta EAD?\)

Hint.

Consider the properties of a regular polygon. What can be inferred about the side lengths of \(\Delta EAD?\)

(c)

Compute \(m\angle{EAD}\text{.}\) How did you determine this?

Hint.

Use the Triangle Angle Sum Theorem to compute \(m\angle{EAD}\text{.}\) Recall that the interior angles of a triangle add up to 180 degrees.

(d)

Determine \(m\angle{DAB}\) showing all work.

Hint.

\(\angle EAB\) is split by segment \(\overline{AD}\) into \(\angle EAD + \angle DAB\text{.}\) Use this to find \(m\angle{DAB}\text{.}\)

(e)

Construct \(\overline{BD}\text{.}\) What is the measure of \(\angle CDE?\) How do you know?

Hint.

Adding \(\overline{BD}\) does not change the measure of \(\angle CDE\text{.}\)

(f)

Describe triangle \(\Delta DAB\) as specifically as possible. Why does this exploration belong in this section?

Hint.

Compare the two diagonals (\(\overline{AD}\) and \(\overline{BD}\)) and the side \(AB\text{.}\) Use the Triangle Angle Sum Theorem.

(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 \(DFCGBHAIEJ\text{.}\)

Hint.

Notice how extending the sides creates new points of intersection outside the pentagon. Consider what shape is created when these points are connected in order.

(ii)

What type of triangle is \(\Delta DFC?\) How is it related to \(\Delta ADB?\)

Hint.

Compare the angles and side relationships in \(\Delta DFC\) to \(\Delta ADB\text{.}\) How are these two triangles related?

Exercises 5.3.5 Exercises

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, \(m\angle ACB=30^{\circ}\) and \(m\angle DEF=108^{\circ}\text{.}\) Answer the following:

Triangle ABC with midpoints DEF as described — Figure 5.3.6.

(a)

Determine \(m\angle DEB\text{.}\)

Hint.

Look at the angles meeting at \(E\text{.}\)

(b)

Determine \(m\angle EFC\text{,}\) \(m\angle BDE\text{,}\) and \(m\angle BAC\text{.}\)

Hint.

Based on the Triangle Angle Sum Theorem, \(m\angle BAC=108\) degrees. Use triangle similarity to find \(m\angle EFC\) and \(m\angle BDE\text{.}\) Hint...Both \(\angle EFC\) and \(\angle BDE\) correspond to \(m\angle BAC\text{.}\)

(c)

Determine \(m\angle FDE\) and \(m\angle AFD\text{.}\)

Hint.

Use angle similarity and match corresponding angles. Which angle in the larger triangle \(\Delta ABC\) corresponds with \(\angle FDE?\) What about \(\angle AFD?\) Use these relationships to find missing measures.

(d)

Determine \(m\angle EFD\text{.}\)

Hint.

The interior angles of any triangle always add up to 180 degrees (Triangle Angle Sum Theorem). For \(\Delta FDE\text{,}\) \(m\angle FDE\) and \(m\angle DEF\) are already known. Use these two angles measures to find \(m\angle EFD\text{.}\)

(e)

Identify four triangles that are similar to \(\Delta ABC\text{.}\) Be sure to list the vertices in order.

Hint.

Imagine \(\Delta ABC\) as the big triangle. If lines are drawn inside it that are parallel to its sides, smaller triangles are made. These smaller triangles will have the same angles as the big one, \(\Delta ABC\text{.}\)

(f)

Determine the length of \(\overline{AF}\text{.}\) Support your answer.

Hint.

Use the definition of a midpoint. What does a midpoint do to a line segment?

(g)

Determine the length of \(\overline{DF}\text{.}\) Support your answer.

Hint.

Line segment \(\overline{DF}\) connects points D and F. Since both are midpoints, the Midsegment Theorem applies. This theorem helps find the length of a line segment that connects two midpoints of a triangle.

(h)

There is a dilation taking \(\Delta ABC\) to \(\Delta FEC\text{.}\) What is the center and scale factor of this dilation?

Hint.

For the center of this dilation, find the vertex that is shared by both \(\Delta ABC\) and \(\Delta FEC\text{.}\) For scale factor, consider how the sides of \(\Delta FEC\) are smaller because F and E are midpoints.

(i)

There is an isometry taking \(\Delta DEB\) to \(\Delta FCE\text{.}\) What type of isometry? Identify its defining object(s).

Hint.

See Table 4.4.12.

(j)

There is an isometry taking \(\Delta ADF\) to \(\Delta EFD\text{.}\) What type of isometry? Identify its defining object(s).

Hint.

See Table 4.4.12. You may want to check orientation as well.

Extending the Concepts

2. Calculating the Golden Ratio.

The ratio \(\frac{1+\sqrt{5}}/2\) that was the constant of proportionality in Exploration 5.3.3 is known as the Golden Ratio.

(a)

Add an algebraic exploration into the number \(\frac{1+\sqrt{5}}{2}\text{,}\) its reciprocal, and its square.

3. Surprising Facts about the Golden Ratio.

The ratio \(\frac{1+\sqrt{5}}{2}\) that was the constant of proportionality in Exploration 5.3.3 is known as the Golden Ratio.

Writing Prompts

4.

Write a letter to a friend in which you use nested similar right triangles, such as in Exploration 5.3.2 to prove the Pythagorean Theorem.

5.

Learn about the Golden Rectangle and the Golden Mean. How are these related to the Golden Triangle?

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