How do you determine the number of possible triangles and find the measure of the three angles given #a=5, b=10, mangleA=30#?

Question

Abigail Armstrong · Accepted Answer

It’s a right triangle with angles measuring #30^@, 60^@, 90^@#

Only one solution possible.

#a = 5, b = 10, hat A = 30^@#

#sin B = (sin 30 * 10) / 5 = 1 or hat B = 90^@#

#hat C = 180 - 30 - 90 = 60^@#

It’s a right triangle with angles measuring #30^@, 60^@, 90^@#

James Adkins · Answer

undefined To determine the number of possible triangles and find the measure of the three angles given the side lengths $ a = 5 $, $ b = 10 $, and $ \angle A = 30^\circ $, you can use the Law of Sines and the properties of triangles.

1. Determine whether the given side lengths can form a triangle by checking the triangle inequality theorem: $ a + b > c $, $ b + c > a $, and $ a + c > b $.

2. If the side lengths satisfy the triangle inequality theorem, calculate the third angle $ \angle C $ using the fact that the sum of the interior angles of a triangle is $ 180^\circ $: $ \angle C = 180^\circ - \angle A - \angle B $.

3. Use the Law of Sines to find the remaining angles $ \angle B $ and $ \angle C $ and any unknown side lengths.

$$ \frac{\sin(\angle A)}{a} = \frac{\sin(\angle B)}{b} = \frac{\sin(\angle C)}{c} $$

$$ \sin(\angle B) = \frac{b \cdot \sin(\angle A)}{a} $$

$$ \sin(\angle C) = \frac{c \cdot \sin(\angle A)}{a} $$

4. Calculate $ \angle B $ and $ \angle C $ using the inverse sine function.

5. Verify if the sum of the angles is $ 180^\circ $.

6. If the side lengths satisfy the triangle inequality theorem and the angles sum up to $ 180^\circ $, then there is one possible triangle. If not, there are no possible triangles with the given measurements.

7. If there is a possible triangle, you now have all three angles and can calculate the third side length if needed.