how many distinct triangles can be constructed?

Question

if a=5, b=7, and angle A= 30 degrees, how many distinct triangles can be constructed?

also,
q#2: if angle A=125 degrees, AB=10, and BC=12, what is the number of distinct triangles that can be constructed?

best answer- 10 pts.

Answer 1

You need to do the sine law to find the missing angle(s). In case 1, sin 30 / 5 = sin B / 7

Since sin 30 = 1/2, cross multiplication gives 3.5 = 5 sin B

So sin B = 3.5/5 = .7, sin^-1 (.7) = 44 degrees or its second quadrant equivalent 180-44 = 136. Since 136 + the given 30 is not over 180, either 44 or 136 will work so there are 2 distinct triangles possible.

This phenomenon is why there is no "SSA" property for congruence of triangles.

For #2, you'd have sin 125 / 12 = sin C / 10

This is 10 sin 125 = 12 sin C, or sin C = (10 sin 125) / 12 = .6826

sin ^-1 .6826 = 43 which added to 125 does not go over 180. But clearly angle C can't be obtuse like in case 1, so here there is only one triangle.

Answer 2

The 5 and 7 long sides have to be next to each other.
The 30-degree angle can be either opposite the 5-side, opposite the 7-side, or opposite the 3rd side. That makes for three possibilities.
If you count mirror images as distinct, then there are six possibilities.

As for the second question, there are only two possibilities (or four, counting mirror images). The 12-side can't be opposite the 125 degree side, since you would then be unable to construct a triangle where another side is length 10.

Answer 3

To make a triangle you need either 2 angles and one side or one angle and two sides. Both answers are only one triangle.

Answer 4

3 triangles

Answer 5

1 distinct triangle for both. Do you want to know the dimensions of each?

Answer 6

Onle ONE!. If a=5, b=7 and Angle A = 30 deg. then only possibilty is ONE trianle.

Answer 7

With the above measurement, we can construct only ONE triangle!