Triangle ABC is given where m<A=33degrees, a=15 in., and the height, h, is 9in.How many distinct triangles can be made with the given measurements? Explain your answer
There are two distinct triangles that can be made with the given measurements.
To determine the number of distinct triangles that can be made, we need to consider the conditions for triangle construction. In a triangle, the sum of the two shorter sides must be greater than the length of the longest side. This is known as the triangle inequality theorem.
Given that side a is 15 inches, the sum of the other two sides, b and c, must be greater than 15 inches for a triangle to be formed. Additionally, the height of the triangle must be less than the length of the corresponding side.
Let's calculate the possible lengths of sides b and c with respect to side a:
b + c > 15
b > 15 - c
Since we are given that h = 9 inches, the possible range of values for side b and c would be:
9 < b < 24
9 < c < 24
Since the sum of the two sides must be greater than 15, the possible values for sides b and c would be:
9 < b < 24
9 < c < 24
b + c > 15
Given these conditions, there are two possible triangles that can be formed:
Triangle 1: Side b = 24 inches, Side c = 24 inches
Triangle 2: Side b = 24 inches, Side c = 16 inches
Hence, there are only two distinct triangles that can be formed with the given measurements while satisfying the triangle inequality theorem.