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.
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
Thank you Reiny!
Is Reiny correct? Please someone tell me.
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.