Determine the value of the unknown variable in each of the following triangles, correct to 1 decimal place.

Question

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For the first triangle, only two angles are given ( $20^\circ$ and $40^\circ$ ). To determine the absolute value of any side length (variables $a, b, c$ ), at least one side length must be provided. Since no side length is given, the unknown variables $a, b, c$ cannot be determined numerically.

For the second triangle:
Given: $\angle A = 33^\circ$ , $\angle C = 65^\circ$ , side $a = BC = 22$ .
Unknown variables: side $b = AC$ and side $c = AB$ .

Calculate the third angle, $\angle B$ .
The sum of angles in a triangle is $180^\circ$ .
$\angle B = 180^\circ - (\angle A + \angle C)$ $\angle B = 180^\circ - (33^\circ + 65^\circ)$ $\angle B = 180^\circ - 98^\circ$ $\angle B = 82^\circ$
Use the Sine Rule to find side $b$ .
The Sine Rule states $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$ .
Using $\frac{b}{\sin B} = \frac{a}{\sin A}$ :
$b = \frac{a \times \sin B}{\sin A}$
Substitute the known values:
$b = \frac{22 \times \sin 82^\circ}{\sin 33^\circ}$ $b \approx \frac{22 \times 0.990268}{0.544639}$ $b \approx \frac{21.7859}{0.544639}$ $b \approx 39.9999$
Rounding to 1 decimal place:
$b \approx 40.0$
Use the Sine Rule to find side $c$ .
Using $\frac{c}{\sin C} = \frac{a}{\sin A}$ :
$c = \frac{a \times \sin C}{\sin A}$
Substitute the known values:
$c = \frac{22 \times \sin 65^\circ}{\sin 33^\circ}$ $c \approx \frac{22 \times 0.906308}{0.544639}$ $c \approx \frac{19.9388}{0.544639}$ $c \approx 36.610$
Rounding to 1 decimal place:
$c \approx 36.6$
The unknown variables in the second triangle are:
$b = \boxed{40.0}$
$c = \boxed{36.6}$

Determine the value of the unknown variable in each of the following triangles, correct to 1 decimal place.

Answer

Step-by-step solution

Calculate the third angle, $\angle B$ .

Use the Sine Rule to find side $b$ .

Use the Sine Rule to find side $c$ .

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More Mathematics Questions

Determine the value of the unknown variable in each of the following triangles, correct to 1 decimal place.

Answer

Step-by-step solution

Calculate the third angle, ∠B\angle B∠B.

Use the Sine Rule to find side bbb.

Use the Sine Rule to find side ccc.

Try one yourself

Still stuck on this one?

More Mathematics Questions

Calculate the third angle, $\angle B$ .

Use the Sine Rule to find side $b$ .

Use the Sine Rule to find side $c$ .