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 our 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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