Question

Solve the right triangle using the information given. Round answers to two decimal places, if necessary a = 7, B = 25°; Find b, c, and A.

Answer 1

b = 3.26

c = 7.72

A = 65°

Explanation:

In right triangle ABC:

• a and b are the legs.
• c is the hypotenuse.

Angles A, B and C are opposite sides a, b and c, respectively. So, angle C is opposite side c (hypotenuse) which means C the right angle and measures 90°.

The interior angles of a triangle sum to 180°. Therefore, given B = 25°:

`A = 180^(\circ) - B - C`

`A = 180^(\circ) - 25^(\circ) - 90^(\circ)`

`A = 65^(\circ)`

To find the length of sides b and c, we can use the sine rule:

`\boxed{\begin{array}{l}\underline{\sf Sine \; Rule} \\\\(a)/(\sin A)=(b)/(\sin B)=(c)/(\sin C)\\\\\\\textsf{where:}\\ \phantom{ww}\bullet \textsf{$A, B$ and $C$ are the angles.}\\ \phantom{ww}\bullet \textsf{$a, b$ and $c$ are the sides opposite the angles.}\\\end{array}}`

Substitute the known values:

`(7)/(\sin 65^(\circ))=(b)/(\sin 25^(\circ))=(c)/(\sin 90^(\circ))`

Therefore, the length of side b is:

`(b)/(\sin 25^(\circ))=(7)/(\sin 65^(\circ))`

`b=(7\sin 25^(\circ))/(\sin 65^(\circ))`

`b=3.264153607...`

`b=3.26\; \sf (2\;d.p.)`

The length of side c is:

`(c)/(\sin 90^(\circ))=(7)/(\sin 65^(\circ))`

`(c)/(1)=(7)/(\sin 65^(\circ))`

`c=(7)/(\sin 65^(\circ))`

`c=7.7236454327...`

`c=7.72\; \sf (2\;d.p.)`

Therefore, the values of b, c and A are:

• b = 3.26
• c = 7.72
• A = 65°