Question

Information is given about ▲ABC . Determine if the information gives one triangle, two triangles, or no triangle. Solve the resulting triangle(s). Round the lengths of the sides and measures of the angles to decimal place if necessary.

a = 48.6, c = 41.7, C = 23°

There ______________ that can be formed from the given information.

Answer 1

Answer:

The correct answer is two triangles.

Step-by-step explanation:

In ▲ABC, a = 48.6, c = 41.7, C = 23° .

Thus this information gives two distinct triangles.

The first triangle is an acute triangle in quadrant 1. The angle measures of this triangle  are ∠A = 27.09° and ∠B = 129.91°. The lengths of the sides of this triangle are a = 48.6; b = 81.86; c = 41.7.

The second triangle is an obtuse triangle in quadrant 2. The angle measures of the other triangle  are ∠A = 152.91° and ∠B = 4.09°. The lengths of the sides of this triangle are a = 48.6; b = 7.61 ; c = 41.7.

Answer 2

In accordance with the law of the cosine, two possible triangles can be constructed.

How to determine the quantity of possible triangles

In this question we must determine the quantity of possible triangles that can be constructed by means of the information given by the law of the sine and the law of the cosine.

By the law of the cosine we determine the length of the side $AC$:

$c^{2} = a^{2}+b^{2}-2\cdot a\cdot b\cdot \cos C$

$41.7^{2} = 48.6^{2}+b^{2}-2\cdot (48.6)\cdot b\cdot \cos 23^{\circ}$

$b^{2}-89.473\cdot b +623.07 = 0$   (1)

There are two solutions: $b_{1}\approx 81.862$, $b_{2}\approx 7.611$. In accordance with the law of the cosine, two possible triangles can be constructed. $\blacksquare$

To learn more on law of the cosine, we kindly invite to check this verified question: https://brainly.com/question/17289163