Determine how many triangles can be drawn. Calculate, then label, all side lengths to the nearest tenth and all interior angles to the nearest degree.

a= 2.1 cm, c= 6.1 cm, <A=20 degrees.

Isn't height c sin A and therefore h=2.1 cm and therefore only 1 90 degree triangle is possible?

sinC/c = sinA/a

sinC = 6.1*0.342/2.1 = 0.993
That is so close to 1, I'd say C=90°

So, yes, you are correct. Just a single right triangle does the trick.

To determine how many triangles can be drawn, we can use the law of sines. The law of sines states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is constant.

Given:
a = 2.1 cm
c = 6.1 cm
<A = 20 degrees

To determine how many triangles can be drawn, we need to find the value of side b. We can use the law of sines:

sin A / a = sin B / b

Rearranging the formula:

b = (a * sin B) / sin A

We have the values for a and A, but we need to find the value of angle B. We can use the fact that the sum of the angles in a triangle is 180 degrees:

B = 180 - A - C

Given <A = 20 degrees, we can find angle C:

C = 180 - A - B

C = 180 - 20 - B

C = 160 - B

Now, we can calculate angle B:

B = 180 - A - C

B = 180 - 20 - (160 - B)

B = 180 - 20 - 160 + B

B + B = 180 - 20 - 160

2B = 180 - 20 - 160

2B = 180 - 180

2B = 0

B = 0 / 2

B = 0 degrees

Since angle B is 0 degrees, it means that the triangle will be degenerate, which means that the triangle will essentially become a straight line.

Therefore, in this case, only one triangle can be drawn, and it will be a degenerate triangle. The sides and angles of the triangle will be as follows:

Side lengths (to the nearest tenth):
a = 2.1 cm
b = 0 cm (degenerate)
c = 6.1 cm

Interior angles (to the nearest degree):
<A = 20 degrees
<B = 0 degrees (degenerate)
<C = 160 degrees

To determine how many triangles can be drawn, we need to use the triangle inequality theorem, which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

In this case, we are given the side lengths a = 2.1 cm and c = 6.1 cm, and the angle A = 20 degrees.

First, let's check if the given side lengths satisfy the triangle inequality theorem:

a + c > c (2.1 + 6.1 > 6.1) => 8.2 > 6.1 (True)
a + c > a (2.1 + 6.1 > 2.1) => 8.2 > 2.1 (True)
a + c > a + c (2.1 + 6.1 > 2.1 + 6.1) => 8.2 > 8.2 (False)

Since the sum of the lengths of any two sides is not greater than the length of the third side (8.2 is not greater than 8.2), we can conclude that no triangle can be formed with the given side lengths.

Therefore, no triangles can be drawn using the given side lengths a = 2.1 cm and c = 6.1 cm, and the angle A = 20 degrees.