# Lesson 5 Special Rights Solidify Understanding

## Learning Focus

Find missing sides of special right triangles without using trigonometry.

Why are

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

### 3.

Draw a

### 4.

Generalize your strategy by letting one side of the triangle measure

Another type of special right triangle is a

### 5.

Draw a

### 6.

Generalize your strategy by letting one side of the triangle measure

### 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:

Anciently, mathematicians, artists and architects were intrigued by the golden ratio—dividing a line segment into two parts

Point

A **golden rectangle** was formed by using the lengths

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

### 2.

Once you have found

## 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

Give the relationship between

## 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.

### 1.

Use

#### a.

Find

#### b.

Find

#### c.

Will

### 2.

#### a.

Identify the type of function.

#### b.

Write the equation of the function.