Question

Two sides of a triangle have measures 3 ft and 6 ft. Also, these sides form a vertex whose angle measures 60 degrees. Calculate the missing attributes of the triangle.

Answer 1

A. The length of the third side is approximately 5 feet.

B. A =
`88^(o)`, B =
`60^(o)` and C =
`32^(o)`.

Explanation:

Let the triangle be ABC. Given two sides and an included angle, let us apply the cosine rule to determine the third length.

A. Let side a = 6 feet and c = 3 feet, thus;

`b^(2)` =
`a^(2)` +
`c^(2)` - 2ac Cos B

=
`6^(2)` +
`3^(2)` - 2(6 x 3) Cos
`60^(o)`

= 36 + 9 - 36 x 0.5

= 45 - 18

`b^(2)` = 27

b =
`√(27)`

= 5.1962

b = 5.2 feet

b ≅ 5 feet

The length of the third side is approximately 5 feet.

B. Given that B =
`60^(o)`, then let us apply the Sine rule to determined the measure of A.

`(a)/(SinA)` =
`(b)/(SinB)`

So that,

`(6)/(Sin A)` =
`(5.2)/(Sin60^(o) )`

Sin A =
`(0.886*6)/(5.2)`

Sin A = 0.99923

A =
`Sin^(-1)` 0.99923

= 87.75

A ≅
`88^(o)`

Since the sum of angle in a triangle =
`180^(o)`.

Then,

A + B + C =
`180^(o)`

`88^(o)` +
`60^(o)` + C =
`180^(o)`

`148^(o)` + C =
`180^(o)`

C =
`180^(o)` -
`148^(o)`

=
`32^(o)`

C =
`32^(o)`

Thus,

A =
`88^(o)`, B =
`60^(o)` and C =
`32^(o)`.