Triangle ABC is given where the measure of angle a is 33°, a=15 in., and the height, h, is 9 in. How many triangles can be made with the given measurements? Explain your answer.

Question

Triangle ABC is given where the measure of angle a is 33°, a=15 in., and the height, h, is 9 in. How many triangles can be made with the given measurements? Explain your answer.

Answer 1

Draw the base AX of the triangle, with AX nice and long.

Draw angle A, making the other side nice and long.
Mark C and drop an altitude to D on AB. Label the height CD=9

Now draw an arc of radius 15 centered at C, and you will see that it intersects AX at two points. Either of those could be labeled B.

Note that 9/AC = sin33°. So, AC = 16.52

Note that if side a had been longer than that, it would have been so long that one of the intersections on AX would have been outside the angle, so there would only be one triangle.

Answer 2

To determine how many triangles can be made with the given measurements, we need to consider the conditions for triangle construction.

In a triangle, the sum of the lengths of any two sides must be greater than the length of the third side. This condition is known as the Triangle Inequality Theorem.

Let's analyze the given measurements:

Side a = 15 in.
Side b = ?
Side c = ?

Angle A = 33°
Angle B = ?
Angle C = ?

Height h = 9 in.

We know that the height of a triangle forms a right angle with the base. In this case, the height is given as 9 inches.

Let's start by finding the missing angle B using the fact that the angles in a triangle add up to 180°:

Angle A + Angle B + Angle C = 180°
33° + Angle B + Angle C = 180°

Since we know Angle A and we want to find Angle B, we can rewrite the equation:

Angle B + Angle C = 180° - Angle A
Angle B + Angle C = 180° - 33°
Angle B + Angle C = 147°

Now let's use the Law of Sines to find the lengths of sides b and c:

sin(A) / a = sin(B) / b = sin(C) / c

We know sin(A) and a:

sin(33°) / 15 = sin(B) / b

To find the value of sin(B), we need to use trigonometric tables or a calculator. Let's assume that we find sin(B) = x.

sin(B) / b = x / b

Since we know sin(B) = x, we can simplify the equation:

x / b = x / b

This means that for any value of x, b can take any positive real number value.

Similarly, we can find the length of side c using the same approach.

Therefore, for the given measurements, there are infinitely many triangles that can be formed. The lengths of sides b and c can vary while satisfying the Triangle Inequality Theorem.