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.
sketch a triangle ABC , where A=15, B=65 and C = 100°
let AB = a and BC = c
by the sine law: c/sin100 = a/sin15
c = asin100/sin15
area of triangle = (1/2)(ac)sin 65
50 = (1/2)(ac)sin65
100 = a(asin100/sin15)(sin65)
a^2 = 100sin15/( sin100sin65)
a^2 = 28.998
a = √28.998 = appr. 5.4
Thank you soooo much!
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.
In this case, we know that the area is 50 square units and we need to find the length of the shortest side.
We can first find the height of the triangle by using the formula:
Area = (1/2) * base * height.
Rearranging the formula, we have:
height = Area / (1/2 * base).
Since we know the area is 50 square units, we can substitute this value into the formula:
height = 50 / (1/2 * base).
Next, we need to find the length of the base. To do this, we can use the Law of Sines, which states that the ratio of the length of a side of a triangle to the sine of the opposite angle is constant.
In this case, we have the angles 15, 65, and 100 degrees, and we want to find the length of the shortest side. Let's call the shortest side b.
Using the Law of Sines:
b / sin(15) = b2 / sin(65),
where b2 is the side opposite the angle of 65 degrees.
Rearranging the equation, we have:
b = (sin(15) / sin(65)) * b2.
Now, we can substitute this value of b into the equation for the height:
height = 50 / (1/2 * base).
Substituting b = (sin(15) / sin(65)) b2, we have:
height = 50 / (1/2 * (sin(15) / sin(65)) * b2).
Simplifying further:
height = 100 / ((sin(15) / sin(65)) * b2).
Now, we need to find the value of b2 to solve for the height.
Using the Law of Sines again, we have:
b2 / sin(100) = b / sin(15),
where b is the side opposite the angle of 15 degrees.
Rearranging the equation, we have:
b2 = (sin(100) / sin(15)) * b.
Substituting this value of b2 back into the equation for the height, we get:
height = 100 / ((sin(15) / sin(65)) * (sin(100) / sin(15)) * b).
Now, we know that height = b * sin(65). So we can find the length of the shortest side b by solving for it in the equation:
b = height / sin(65).
Now we have all the information needed to find the length of the shortest side. Plug in the values and solve for b:
b = (100 / ((sin(15) / sin(65)) * (sin(100) / sin(15)))) / sin(65).
Using a calculator or software that can compute trigonometric functions, we can find the value of b.