Question

How many triangles can be formed if A = 82°, a = 21 cm, and b = 16 cm?

Answer 1

If you have atriangle with sides a, b and c and internal angles A, B and C

One side of a triangle is less than the sum of the other two sides and greather than their difference:

A=82º

As this is an obtuse triangle, the angle A is the angle with greater measure, then the opposite side of this angle will be the side with greater measure.

The opposite side to angle A is a, then:

`5The internal angles of a triangle sum 180º:[tex]\begin{gathered} 82º+B+C=180º \\ B+C=180-82 \\ B+C=98º \end{gathered}`

The relation between the angles and the sides in a triangle:

`\begin{gathered} (a)/(\sin A)=(b)/(\sin B)=(c)/(\sin C) \\ \\ (21)/(\sin82)=(16)/(\sin B) \\ 21\sin B=16\sin 82 \\ \sin B=(16)/(21)\sin 82 \\ B=\sin ^(-1)((16)/(21)\sin 82) \\ B=48.98º \end{gathered}`

Angle B is 48.98º

`\begin{gathered} C=98-48.98 \\ C=49.02º \end{gathered}`
`\begin{gathered} (16)/(\sin48.98)=(c)/(\sin 49.02) \\ \\ (16\sin 49.12)/(\sin 48.98)=c \\ \\ c=16.03 \end{gathered}`

As you can see the given data allow you to find an exact measure for the other side and angles in the triangle.

Then, as you can see, you can just form one triangle with A=82º, a=21cm b=16cm