Determine the number of triangles that could be drawn with the given measure. Then, find the measures of the other angles and the other side in each possible triangle. Round side lengths to the nearest tenth of a unit and angle measure to the nearest tenth of a degree, where necessary. a) triangle ABC, where angle A=125', a=3m, b=5m.
Basic property of triangles:
the largest side is opposite the largest angle
the smallest side is opposite the smallest angle
etc
since angle A is 125° , the sum of the other two angles must be 55°
that is, "a" must be the largest side, but b > a
which would mean angle B > angle A
But we can't have 2 obtuse angles, so ....
no such triangle is possible
ok thanks a lot for your help :D
To determine the number of triangles that could be drawn with the given measures, we'll use the triangle inequality theorem. According to the theorem, for a triangle with side lengths a, b, and c, the sum of any two sides must be greater than the third side. Let's apply this theorem to the given triangle.
Triangle ABC:
Angle A = 125°
Side a = 3m
Side b = 5m
To find the number of triangles that can be formed, we need to check if the sum of any two sides is greater than the third side. Let's do this with side lengths a, b, and c.
a + b > c
3 + 5 > c
8 > c
Since we only have the side lengths a and b, and they are already greater than c, we don't need to check further. This means that a triangle can be formed with these measures.
To find the measures of the other angles and the other side of the triangle, we can use the law of sines and the law of cosines.
Using the law of sines:
sin(A)/a = sin(B)/b = sin(C)/c
We know angle A = 125° and side a = 3m, so we can rearrange the formula to find sin(B):
sin(B) = (sin(A) * b) / a
Substituting the values:
sin(B) = (sin(125°) * 5) / 3
Using a calculator, we find that sin(B) ≈ 0.8717.
Now, we can find angle B using the inverse sine function:
B = sin^(-1)(0.8717)
Using a calculator, we find that angle B ≈ 61.6°.
To find angle C, we can use the fact that the sum of angles in a triangle is 180°:
C = 180° - A - B
C = 180° - 125° - 61.6°
Using a calculator, we find that angle C ≈ -6.6°.
However, a triangle cannot have a negative angle, so the angle C is invalid. This means that the measures given for the triangle ABC are not possible.