Lesson 11 Special Rights Solidify Understanding

Learning Focus

Find missing sides of special right triangles without using trigonometry.

Why are $30 ° - 60 ° - 90 °$ and $45 ° - 45 ° - 90 °$ triangles considered to be “special?”

Why can we find the missing sides of these right triangles without using trigonometry?

Are there any other special right triangles?

Open Up the Math: Launch, Explore, Discuss

The Pythagorean theorem and right triangle trigonometry are both useful mathematical tools when trying to find missing sides of a right triangle.

1.

What do you need to know about a right triangle in order to use the Pythagorean theorem?

2.

What do you need to know about a right triangle in order to use right triangle trigonometry?

While using the Pythagorean theorem is fairly straightforward (you only have to keep track of the legs and hypotenuse of the triangle), right triangle trigonometry generally requires a calculator to look up values of different trigonometric ratios. There are some right triangles, however, for which knowing a side length and an angle is enough to calculate the value of the other sides without using trigonometry. These are known as special right triangles because their side lengths can be found by relating them to another geometric figure for which we know something about its sides.

One type of special right triangle is a $45 ° - 45 ° - 90 °$ triangle.

3.

Draw a $45 ° - 45 ° - 90 °$ triangle and assign a specific value to one of its sides. (For example, let one of the legs measure $8 cm$ , or choose to let the hypotenuse measure $5 inches$ . You will want to try both approaches to perfect your strategy.) Now that you have assigned a measurement to one of the sides of your triangle, find a way to calculate the measures of the other two sides. As part of your strategy, you may want to relate this triangle to another geometric figure that may be easier to think about.

4.

Generalize your strategy by letting one side of the triangle measure $x$ . Show how the measure of the other two sides can be represented in terms of $x$ . (Make sure to consider cases where $x$ is the length of a leg, as well as the case where $x$ is the length of the hypotenuse.)

Another type of special right triangle is a $30 ° - 60 ° - 90 °$ triangle.

5.

Draw a $30 ° - 60 ° - 90 °$ triangle and assign a specific value to one of its sides. Now that you have assigned a measurement to one of the sides of your triangle, find a way to calculate the measures of the other two sides. As part of your strategy, you may want to relate this triangle to another geometric figure that may be easier to think about.

6.

7.

Can you think of any other angle measurements that will create a special right triangle?

Ready for More?

An interesting “special” right triangle was discovered by the astronomer and mathematician Johannes Kepler (1571–1630). The Kepler Right Triangle has side lengths that form three terms in a geometric sequence: $a$ , $a r$ , $a r^{2}$ , where $a$ is the length of the shorter leg of the right triangle and $r$ is the “golden ratio.”

Anciently, mathematicians, artists and architects were intrigued by the golden ratio—dividing a line segment into two parts $m$ and $n$ so that the ratio of the longer part $m$ to the shorter part $n$ was equivalent to the ratio of the entire length of the line segment $m + n$ to the longer part $m$ , that is, $\frac{m}{n} = \frac{m + n}{m}$ .

Point $C$ divides line segment $A B$ in a golden ratio if $\frac{m}{n} = \frac{m + n}{m}$ . That is, $n$ , $m$ , $n + m$ form a geometric sequence: multiplying length $n$ by the golden ratio gives length $m$ , and multiplying length $m$ by the golden ratio gives us length $m + n$ .

A golden rectangle was formed by using the lengths $m$ and $m + n$ as the sides, forming an $m$ by $m + n$ rectangle with area $m^{2} + m n$ . An interesting feature of a golden rectangle is that when the square with area $m^{2}$ is removed from the rectangle, the remaining $m$ by $n$ rectangle is also a golden rectangle. Removing another square from this rectangle produces another golden rectangle, and so on ...

Kepler was fascinated to find that the ratio of the hypotenuse to short leg in a Kepler Triangle was also the golden ratio.

Here is your task:

1.

Find the exact value of the golden ratio using the information given above. For simplicity, let the shorter part $n = 1$ so that $m = r$ , the golden ratio. (Hint: Write and solve a quadratic equation that is equivalent to the proportionality statement.)

2.

Once you have found $r$ , show that 1, $r$ , $1 + r$ forms a geometric sequence. If they do, then these lengths will form a Kepler Right Triangle.

Takeaways

Sometimes we don’t need to use trigonometry to find missing sides of a right triangle when only the angles and one side length is known. Triangles for which this is possible are called special right triangles.

Give the relationship between $x$ and the other sides of the right triangle in each of the following $45 ° - 45 ° - 90 °$ triangles:

Give the relationship between $x$ and the other sides of the right triangle in each of the following $30 ° - 60 ° - 90 °$ triangles:

Vocabulary

special right triangles
Bold terms are new in this lesson.

Lesson Summary

In this lesson, we learned there are some special right triangles for which missing sides of the triangle can be found when only one side is known, without using trigonometry! This happens when the right triangle is the result of decomposing a familiar shape, such as a square or an equilateral triangle, into two congruent right triangles.

Retrieval

1.

Use $△ B C D$ to answer the problems. Round answers to hundredths place.

$G i v e n : m ∠ C B A = 42 °$ and $m ∠ C D A = 35 °$ , and $A C = 22 in.$

a.

Find $m ∠ B C D$ , $m ∠ B C A$ , and $m ∠ A C D$ .

b.

Find $B C$ and $C D$ .

c.

Will $B D$ be longer or shorter than $B C + C D$ ? Why?

2.

a.

Identify the type of function.

b.

Write the equation of the function.