PROBLEMS ON SPECIAL RIGHT TRIANGLES

Special right triangles are the triangles whose angle measures are

45° - 45° - 90° (or) 30° - 60° - 90°

45° - 45° - 90° triangle theorem :

In a 45° - 45° - 90° triangle, the hypotenuse is √2 times as long as each length. 

30˚- 60˚- 90˚ triangle theorem :

In a 30° - 60° - 90° triangle, the hypotenuse is twice as long as the shorter length, and the longer length is √3 times as long as the shorter length.  

Example 1 :

Solution  :

Given triangle is a 30˚-60˚-90˚ triangle.

Finding the value of a :

By 30˚-60˚-90˚ triangle theorem,

Hypotenuse  =  2 . shorter length

Here hypotenuse  =  a, and shorter length  =  5

a  =  2 . 5

a  =  10

So, the value of a is 10.

Finding the value of b :

By 30˚-60˚-90˚ triangle theorem,

Longer length  =  √3 . shorter length

 Here longer length  =  b, and shorter length  =  5

b  =  √3 . 5

b  =  5√3

So, the value of b is 5√3.

Example 2 :

Solution  :

Given triangle is a 45˚-45˚-90˚ triangle.

Finding the value of a :

Since the given triangle has base angles that are equal, the two sides' lengths are equal.

a  =  8

So, the value of a is 8.

Finding the value of b :

By 45˚-45˚-90˚ triangle theorem,

Hypotenuse  =  √2 . length

Here hypotenuse  =  b, and length  =  8

=  √2 . 8

b  =  8√2

So, the value of b is 8√2.

Example 3 :

Solution :

Given triangle is a 30˚-60˚-90˚ triangle.

Finding the value of a :

By 30˚-60˚-90˚ triangle theorem,

Hypotenuse  =  2 . shorter length

 Here hypotenuse  =  12, and shorter length  =  a

12  =  2 . a

a  =  6

So, the value of a is 6.

Finding the value of b :

By 30˚-60˚-90˚ triangle theorem,

Longer length  =  √3 . shorter length

 Here longer length  =  b, and shorter length(a)  =  6

b  =  √3 . 6

b  =  6√3

So, the value of b is 6√3.

Example 4 :

Solution :

Given triangle is a 45˚-45˚-90˚ triangle.

Finding the value of a :

By 45˚-45˚-90˚ triangle theorem,

Hypotenuse  =  √2 . length

Here hypotenuse  =  3√2, and length  =  a

3√2  =  √2 . a

3√2/√2  =  a

a  =  3

So, the value of a is 3.

Finding the value of b :

Since the given triangle has base angles that are equal, the two sides' lengths are equal.

b  =  3

So, the value of b is 3.

Example 5 :

Solution :

Given triangle is a 45˚-45˚-90˚ triangle.

Finding the value of a :

Since the given triangle has base angles that are equal, the two sides' lengths are equal.

a  =  2√2

So, the value of a is 2√2.

Finding the value of b :

By 45˚-45˚-90˚ triangle theorem,

Hypotenuse  =  √2 . length

Here hypotenuse  =  b, and length  =  2√2

b  =  √2 . 2√2

b  =  2 . 2

b  =  4

So, the value of b is 4.

Example 6 :

Solution :

Given triangle is a 30˚-60˚-90˚ triangle.

Finding the value of a :

By 30˚-60˚-90˚ triangle theorem,

Hypotenuse  =  2 . shorter length

 Here hypotenuse  =  14, and shorter length  =  a

14  =  2 . a

a  =  7

So, the value of a is 7.

Example 7 :

Solution  :

Given triangle is a 30˚-60˚-90˚ triangle.

Finding the value of a :

By 30˚-60˚-90˚ triangle theorem,

Hypotenuse  =  2 . shorter length

 Here hypotenuse  =  a, and shorter length  =  6

a  =  2 . 6

a  =  12

So, the value of a is 12.

Finding the value of b :

By 30˚-60˚-90˚ triangle theorem,

Longer length  =  √3 . shorter length

 Here longer length  =  b, and shorter length  =  6

b  =  √3 . 6

b  =  6√3

So, the value of b is 6√3

Example 8 :

Solution :

Given triangle is a 45˚-45˚-90˚ triangle.

Finding the value of a :

By 45˚-45˚-90˚ triangle theorem,

Hypotenuse  =  √2 . length

Here hypotenuse  =  16, and length  =  a

16  =  √2 . a

a  =  16/√2

To rationalize denominator in 16/√2, multiply numerator and denominator  by √2. 

a  =  (16/√2) . (√2/√2)

a  =  16√2/2

a  =  8√2

So, the value of a is 8√2.

Finding the value of b :

Since the given triangle has base angles that are equal, the two sides' lengths are equal.

b  =  8√2

So, the value of b is 8√2.

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