Special Right Triangle Worksheet

**Page 1**

1.

Find the value of $y$ if $a$ = 10 cm .

a. | 20 $\sqrt{3}$ cm | ||

b. | 10 $\sqrt{3}$ cm | ||

c. | 20 cm | ||

d. | 10 cm |

[30

[Substitute.]

[Multiply.]

Correct answer : (3)

2.

The sides of a triangle are 5 in., 5 in., 5$\sqrt{2}$ in. respectively. The measure of the greatest angle is

a. | 90 ^{o} | ||

b. | 150 ^{o} | ||

c. | 120 ^{o} | ||

d. | 60 ^{o} |

As the converse of the Pythagoras theorem holds good, the triangle is right angled.

In a right triangle, the maximum angle is 90

Correct answer : (1)

3.

The sides of a triangle are 8 cm, 8$\sqrt{3}$ cm and 16 cm respectively. The measure of the least angle is

a. | 30 ^{o} | ||

b. | 60 ^{o} | ||

c. | 45 ^{o} | ||

d. | 90 ^{o} |

As the converse of the Pythagorean theorem holds good, the triangle is right angled.

8 : 8

As the ratio of sides is 1 :

[30

The minimum angle is 30

Correct answer : (1)

4.

For a given perimeter, which of the following will have maximum area?

a. | 60 ^{o}-60^{o}-60^{o} triangle | ||

b. | 30 ^{o}- 30^{o}-90^{o} triangle | ||

c. | 50 ^{o}-60^{o}-70^{o} triangle | ||

d. | 45 ^{o}-45^{o}-90^{o} triangle |

As an equilateral triangle is most symmetrical, it will have the maximum area.

A '60

Correct answer : (1)

5.

Find the length of $\stackrel{\u203e}{\mathrm{QR}}$. [Given RS = 4 units.]

a. | 2$(\sqrt{3}-1)$ units | ||

b. | 2$(\sqrt{3}+1)$ units | ||

c. | $\frac{\sqrt{3}-1}{2}$ units | ||

d. | 2$\sqrt{3}$ units |

[ΔPQR is an isosceles right triangle.]

[Tan 30.]

[Step 1.]

Þ

[Step 1.]

QR =

[Simplify.]

QR = 2

[Simplify.]

Correct answer : (2)

6.

The lengths of the three sides of different triangles are given. Select the isosceles right triangles.

1. 3, 4, and 5 units each

2. 4, 4, and 4√2 units each

3. 4, 4, and 7 units each

4. 2√3, 2√3, and 2√6 units each

1. 3, 4, and 5 units each

2. 4, 4, and 4√2 units each

3. 4, 4, and 7 units each

4. 2√3, 2√3, and 2√6 units each

a. | 1 only | ||

b. | 3 only | ||

c. | 2 and 4 only | ||

d. | 2, 3, and 4 only |

In right triangle, if lengths of two sides are equal then it is an isosceles right triangle.

Every isosceles triangle need not be an isosceles right triangle.

Measures 3, 4 and 5 units represent a right triangle, but it is not isosceles.

Measures 4, 4, and 4√2 represent an isosceles triangle which is also a right triangle.

Measures 4, 4, and 7 represent an isosceles triangle but it is not a right triangle.

Measures 2√3, 2√3, and 2√6 represent an isosceles triangle which is also a right triangle.

So, only measures in 2 and 4 represent isosceles right triangles.

Correct answer : (3)

7.

Is the ratio of the length of legs same as the ratio of their opposite angles in an isosceles right triangle?

a. | No | ||

b. | Yes |

The side opposite to right angle is hypotenuse and the remaining sides are legs.

In a 45

In a 45

The ratio of the legs = 1 : 1

Opposite angles of the legs are 45

The ratio of the angles = 45 : 45 = 1 : 1

In an isosceles right triangle, the ratio of the length of legs is same as the ratio of their opposite angles.

Correct answer : (2)

8.

Is a triangle with side lengths 7, 14 and 7$\sqrt{3}$ a 30^{o}-60^{o}-90^{o} triangle?

a. | yes | ||

b. | no, it is not a right triangle | ||

c. | no, it is an equilateral triangle | ||

d. | no, it is a 45 ^{o}- 45^{o}- 90^{o} triangle |

14

[Checking for Pythagorean theorem.]

Since given measures satisfy Pythagorean theorem, they form a right triangle.

Here, the length of hypotenuse is 14 and the length of shorter leg is 7.

Since the length of hypotenuse is twice the length of shorter leg, the sides form a 30

Correct answer : (1)

9.

What are the angles of an isosceles right triangle?

a. | 30 ^{o}, 60^{o} and 90^{o} | ||

b. | 45 ^{o}, 45^{o}and 90^{o} | ||

c. | 40 ^{o}, 40^{o} and 100^{o} | ||

d. | 30 ^{o}, 30^{o} and 90^{o} |

Sum of angles in a triangle = 180

In right triangle one angle is 90

Sum of the other two angles is 90

Since other two angles are equal, each angle = 45

[Since

The angles in a right isosceles triangle are 45

Correct answer : (2)

10.

What is the ratio of the length of the hypotenuse to the length of the leg of an isosceles right triangle?

a. | 2 : 3 | ||

b. | 1 : $\sqrt{2}$ | ||

c. | $\sqrt{2}$ : 1 | ||

d. | 3 : 2 |

Length of hypotenuse =

The ratio of the length of the hypotenuse to the length of leg in an isosceles right triangle is

Correct answer : (3)