Question

Triangle ABC is given where m(angle)A=33 degrees, a=15 in., and the height, h, is 9 in. How many distinct triangles can be made with the given measurements? Explain your answer.

Answer 1

This is referred to as the "ambigious case"

From the given:
9/AC = sin33
AC = 9/sin33 = appr 16.5

now by the sine law sin(angle ABC) /16.5 = sin33/15
sin(angle ABC) = .6
angle ABC = 36.9° or angle ABC = 180-36.9 = 143.1°

So, two angles
both triangles can be drawn

Answer 2

Thank you Reiny!

Answer 3

Is Reiny correct? Please someone tell me.

Answer 4

[AI-generated answer — Explain Bot]

To determine the number of distinct triangles that can be made with the given measurements, we can use the inequality theorem.

The inequality theorem states that for a triangle with side lengths a, b, and c, the sum of any two sides must be greater than the third side. In mathematical terms, this can be written as:
a + b > c
a + c > b
b + c > a

In this case, we are given that side a is 15 in., and the height, h, is 9 in. The height, also known as an altitude, is perpendicular to side a and divides the triangle into two right-angled triangles.

From the given information, we can deduce that side b is the base of the right-angled triangles and side c is the hypotenuse.

Using trigonometry, we know that the sine of an angle is equal to the ratio of the opposite side to the hypotenuse:
sin(A) = h/c

Let's substitute the values we know into this equation:
sin(33) = 9/c

Simplifying this equation will give us the value of c:
c = 9 / sin(33)

Now that we know the lengths of sides a, b, and c, we can apply the triangle inequality theorem to determine if these lengths are valid for forming a triangle:

a + b > c
15 + b > 9 / sin(33)

To find the range of values for side b, we isolate it:
b > 9 / sin(33) - 15

Therefore, any value of b greater than 9 / sin(33) - 15 will satisfy the triangle inequality theorem and form a valid triangle.

Hence, there are an infinite number of distinct triangles that can be made with the given measurements, as there are an infinite number of values for side b that satisfy the inequality theorem.