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# Special Right Triangle Worksheet

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

#### Solution:

Hypotenuse = 2 × shorter leg
[30o- 60o- 90o triangle theorem.]

y = 2 × 10
[Substitute.]

y = 20 cm
[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. 90o b. 150o c. 120o d. 60o

#### Solution:

52 + 52 = (52)2

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

In a right triangle, the maximum angle is 90o.

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. 90o

#### Solution:

82 + (83)² = 16²

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

8 : 83 : 16 = 1 : 3 : 2.

As the ratio of sides is 1 : 3 : 2, the angles are 30o, 60o, 90o
[30o-60o-90o triangle theorem.]

The minimum angle is 30o.

Correct answer : (1)
4.
For a given perimeter, which of the following will have maximum area?
 a. 60 o-60o-60o triangle b. 30 o- 30o-90o triangle c. 50 o-60o-70o triangle d. 45 o-45o-90o triangle

#### Solution:

For a given perimeter, the triangle with greater symmetry will have greater area.

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

A '60 o-60o-60o triangle' represents an equilateral triangle. So, this triangle will have maximum area.

Correct answer : (1)
5.
Find the length of $\stackrel{‾}{\mathrm{QR}}$. [Given RS = 4 units.]

 a. 2$\left(\sqrt{3}-1\right)$ units b. 2$\left(\sqrt{3}+1\right)$ units c. $\frac{\sqrt{3}-1}{2}$ units d. 2$\sqrt{3}$ units

#### Solution:

Length of PQ = length of QR
[ΔPQR is an isosceles right triangle.]

PQ / QS =13
[Tan 30.]

QR / QR + RS =13
[Step 1.]

Þ QR / QR + 4 = 13
[Step 1.]

3×QR = QR + 4

(3- 1)×QR =  4

QR = 43-1
[Simplify.]

QR = 2(3+1) units
[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
 a. 1 only b. 3 only c. 2 and 4 only d. 2, 3, and 4 only

#### Solution:

If the lengths of two sides are equal then it is an isosceles triangle.

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

#### Solution:

In an isosceles right triangle, the angles are 45o - 45o - 90o.

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

In a 45o - 45o - 90o triangle, the sides opposite to 45o angles are the legs.

In a 45o - 45o - 90o triangle, the two legs are same in length.

The ratio of the legs = 1 : 1

Opposite angles of the legs are 45o and 45o.

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 30o-60o-90o triangle?
 a. yes b. no, it is not a right triangle c. no, it is an equilateral triangle d. no, it is a 45o- 45o- 90o triangle

#### Solution:

The side lengths of the triangle are 7, 14 and 73.

142 = 72 + (73) 2
[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 30o - 60o - 90o triangle.

Correct answer : (1)
9.
What are the angles of an isosceles right triangle?
 a. 30o, 60o and 90o b. 45o, 45oand 90o c. 40o, 40o and 100o d. 30o, 30o and 90o

#### Solution:

If two angles of a right triangle are same then the triangle is said to be an isosceles right triangle.

Sum of angles in a triangle = 180o

In right triangle one angle is 90o.

Sum of the other two angles is 90o.

Since other two angles are equal, each angle = 45o.
[Since 90°2= 45o.]

The angles in a right isosceles triangle are 45o, 45o and 90o.

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

#### Solution:

In an isosceles right triangle, hypotenuse is 2 times the length of a leg.

Length of hypotenuse = 2 × length of a leg.

Length of hypotenuselength of leg = 21

The ratio of the length of the hypotenuse to the length of leg in an isosceles right triangle is 2 : 1.

Correct answer : (3)

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