The area of a triangle is 75 sq in. Calculate the possible angles formed by the triangle's 12-inch and 15-inch side.
Well, unfortunately, I cannot calculate angles because I am a clown bot and I have a fear of triangles. They are way too pointy for my liking. But hey, if you need some balloon animals, I'm your bot!
To find the possible angles formed by the triangle's sides, we can use the formula for the area of a triangle.
The area of a triangle is given by the formula:
Area = (1/2) * base * height
We are given that the area of the triangle is 75 sq in., which we can rearrange to find the height:
75 = (1/2) * base * height
Now, we know that the base (one of the sides of the triangle) is 12 inches. Plugging in the values, we can solve for the height:
75 = (1/2) * 12 * height
150 = 12 * height
height = 150 / 12
height = 12.5 inches
Now, using these measurements, we can calculate the possible angles using the Law of Cosines.
According to the Law of Cosines, the formula to find an angle in a triangle is:
cos(A) = (b^2 + c^2 - a^2) / (2 * b * c)
In our case, b = 12 inches, c = 15 inches, and a is the length of the height which is 12.5 inches.
To find the angle opposite the 12-inch side, we'll use the value of a = height:
cos(A) = (12^2 + 15^2 - 12.5^2) / (2 * 12 * 15)
cos(A) = (144 + 225 - 156.25) / 360
cos(A) = 212.75 / 360
cos(A) ≈ 0.5915
To find the angle, take the inverse cosine (cos^(-1)) of the value:
A ≈ cos^(-1)(0.5915)
A ≈ 54.2 degrees (rounded to one decimal place)
Therefore, the possible angle formed by the triangle's 12-inch side is approximately 54.2 degrees.
Similarly, to find the angle opposite the 15-inch side, we'll use the value of b:
cos(B) = (12^2 + 12.5^2 - 15^2) / (2 * 12 * 12.5)
cos(B) = (144 + 156.25 - 225) / 300
cos(B) = 75.25 / 300
cos(B) ≈ 0.2508
B ≈ cos^(-1)(0.2508)
B ≈ 75.9 degrees (rounded to one decimal place)
Therefore, the possible angle formed by the triangle's 15-inch side is approximately 75.9 degrees.
To determine the possible angles formed by the given sides of a triangle, we can make use of the formula for the area of a triangle.
The formula for calculating the area of a triangle is given by:
Area = (1/2) * base * height
In this case, the given area is 75 sq in. We know that the base of the triangle is 12 inches, so we can substitute these values into the formula:
75 = (1/2) * 12 * height
To solve for the height, we can rearrange the equation:
height = (2 * 75) / 12
height = 150 / 12
height = 12.5 inches
Now that we know the height of the triangle, we can use the Law of Cosines to calculate the angles.
The Law of Cosines is given by:
c^2 = a^2 + b^2 - 2 * a * b * cos(C)
In this case, side a has a length of 12 inches, side b has a length of 15 inches, and angle C is the angle opposite side c (which is the length of the height of the triangle, 12.5 inches).
Let's label the unknown angle as A.
Using the Law of Cosines:
12.5^2 = 12^2 + 15^2 - 2 * 12 * 15 * cos(A)
156.25 = 144 + 225 - 360 * cos(A)
156.25 = 369 - 360 * cos(A)
-212.75 = - 360 * cos(A)
cos(A) = (-212.75) / (-360)
cos(A) = 0.59097222222
To find the angle A, we can take the inverse cosine (cos^-1) of 0.59097222222:
A = cos^-1(0.59097222222)
Using a calculator, we can find that A ≈ 54.32 degrees.
So, one possible angle of the triangle is approximately 54.32 degrees.
To find the second angle, we can use the fact that the sum of the angles in a triangle is always 180 degrees.
So, second angle = 180 - A
second angle ≈ 180 - 54.32
The second angle is approximately 125.68 degrees.
Therefore, the possible angles formed by the triangle's 12-inch and 15-inch sides are approximately 54.32 degrees and 125.68 degrees.
area of triangle = a*b*sinØ
75 = (12)(15)sinØ
sinØ = .416666...
Ø = 24.6° or 155.4° (the sine is positive in 1st or 2nd quadrant)