If a triangle has an area of 50 square units and has angles that measure 15, 65, and 100 degress, find the length of the shortest side to the nearest tenth. Do not round till the final answer.
To find the length of the shortest side of the triangle, we can use the formula for the area of a triangle:
Area = (1/2) * base * height
First, let's find the height of the triangle. We know that the base of the triangle is one of its sides, while the height is perpendicular to the base. Since we don't have the length of the base or the height, we need to find one of them.
To find the height, we need to use trigonometry. The side opposite the smallest angle (15 degrees) is the height of the triangle. The side opposite the largest angle (100 degrees) is the base of the triangle. We can use the sine function to find the height:
sin(15 degrees) = height / side opposite 15 degrees
Next, we can rearrange the equation to solve for the height:
height = side opposite 15 degrees * sin(15 degrees)
Now, let's find the side opposite 15 degrees. Since we don't have any side lengths given, it's helpful to give a variable name to the side opposite 15 degrees. Let's call it "x".
Using the sine function again, we can write the equation:
sin(65 degrees) = x / side opposite 65 degrees
Rearranging the equation to solve for x:
x = side opposite 65 degrees * sin(65 degrees)
Now we have expressions for both the height and the side opposite 15 degrees in terms of unknown variables. We can substitute these expressions into the area formula to solve for the length of the shortest side:
Area = (1/2) * base * height
50 = (1/2) * x * (side opposite 15 degrees * sin(15 degrees))
Simplifying the equation:
100 = x * (side opposite 15 degrees * sin(15 degrees))
Since we know that the sum of the angles in a triangle is always 180 degrees, we can find the measure of the remaining angle:
180 - 15 - 65 = 100 degrees
Therefore, the side opposite 15 degrees is also the side opposite 100 degrees.
Now, we can substitute the expression for the side opposite 15 degrees with "x" and solve for x:
100 = x * (x * sin(15 degrees))
Dividing both sides of the equation by sin(15 degrees):
100 / sin(15 degrees) = x^2
Taking the square root of both sides:
sqrt(100 / sin(15 degrees)) = x
Now we have the value of x, which represents the length of the shortest side of the triangle. To find its approximate value to the nearest tenth, we can substitute the value of sin(15 degrees) and calculate x using a calculator:
x ≈ sqrt(100 / 0.258819) ≈ 25.7
Therefore, the length of the shortest side of the triangle is approximately 25.7 units to the nearest tenth.