Two Special Right Triangles |

**Two Special Right Triangles**

There are two right-angled triangles which occur frequently in our work in geometry, analytic geometry and trigonometry, which we should learn to make certain types of problems easier to solve. They are the **isosceles right-angled triangle** and **the 30°, 60° , 90° triangl**e. Using the **Pythagorean** theorem, we can determine **the ratios of the sides** of these triangles.

**The Isosceles Right Triangle**

The ratio of the sides is 1 : 1 : .

So, if the equal sides measure 5 cm each, the hypotenuse is cm long.

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**Examples:**

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**The 30°, 60°, 90° Triangle**

The ratio of the lengths of the sides in this triangle are .

**Note:** If in a right triangle, the hypotenuse is twice a side, it is a 30°, 60°, 90° triangle since the ratio of the hypotenuse to the side opposite the 30° is 2 : 1

**Example:**

A sculpture is composed of 2 similar right circular cones. The measures are as shown.

Find the total height of the sculpture to the nearest centimeter.

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Now get a pencil, an eraser and a note book, copy the questions,

do the practice exercise(s), then check your work with the solutions.

If you get stuck, review the examples in the lesson, then try again.

**Practice**

1/ The diagram shows the side view of a roof.

a) find BA or h, the height of the roof |

b) find DC, the base of the roof |

c) find the area of the side of the roof. |

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2/ In the diagram, *AB // DE* and the measures are as shown.

- a) What is the measure of angle

- b) How long is

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3/ In the diagram, the measures are as shown:

- Find the lengths of all the sides and the measures of all the acute angles.

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**Solutions**

1/ a) Since *BA *or *h* is opposite the 30º, it = ½ the hypotenuse. *h* = ½(4.8)* *= 2.4 or 12/5 cm*.*

b) *DC* is * DA + AC*. *(see diagram)*

Since *AC* is opposite the 60º

Since *DA* is opposite the 30º, *DA = BA* ÷ root 3.

*DC* is * DA + AC* =

c) The area of the roof ½ (base)(height) =

2/ a) Because *BC = 2 BA*, and angle *A = 90**º*, angle BCA = 30º

So, this is the *30º, 60º, 90º* triangle with *angle B = 60º*

Since *BA // DE*, angle *CDE **= 60º*.

b)

3/

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