If a triangle has angle measures 15, 120 and 45, and the length of the side between the 15 and 120 angles is 8, what is the area of the triangle?

To find the area of the triangle, we can use the formula:

Area = (1/2) * base * height

First, let's find the height of the triangle. To do this, we need to find the length of the altitude from the vertex of the 120-degree angle to the base. Since this is an isosceles triangle (with two equal sides), the altitude will bisect the base.

Let's consider the triangle with angle measures 15, 120, and 45.

Step 1: Draw a triangle with angle measures 15, 120, and 45.

/\
/_\
/ \
/ \
/ \
/_________\
/ 8 \

Step 2: Draw the altitude from the vertex of the 120-degree angle to the base.

/\
/_\
/ | \
/ | \
/ | h \
/_____|____\
/ 8 \

Step 3: Split the triangle into two right triangles.

/\
/_\
/ ++\
/ + | \
h1 / + | \ h2
/___|______\
/ 8 \

Step 4: Find the length of the hypotenuse of h1.

The angle measures of h1 are 15, 75, and 90 degrees. Since the triangle is not an angle we typically recognize, we'll need to use trigonometry.

Since we know the adjacent side (8), we can use the tangent function to find h1.

tan(15) = opposite / adjacent
tan(15) = h1 / 8

Rearranging the equation, we get:
h1 = 8 * tan(15)

Step 5: Find the length of the hypotenuse of h2.

The angle measures of h2 are 45, 45, and 90 degrees. Since this is a special right triangle (an isosceles right triangle), we can use the Pythagorean theorem.

Since we know the length of one leg (8), we can find the length of the hypotenuse (h2) using the theorem:
a^2 + b^2 = c^2

8^2 + 8^2 = h2^2
64 + 64 = h2^2
128 = h2^2
h2 = sqrt(128)

Step 6: Find the height (h) of the triangle by adding h1 and h2.

h = h1 + h2
h = 8 * tan(15) + sqrt(128)

Step 7: Find the length of the base.

Since h bisects the base, we can find the length of the base by doubling the length between the 15 and 120-degree angles.

base = 2 * 8
base = 16

Step 8: Calculate the area using the formula.

Area = (1/2) * base * height
Area = (1/2) * 16 * [8 * tan(15) + sqrt(128)]

Now you can substitute the values into the formula to calculate the area.

To find the area of a triangle, we can use the formula A = (1/2) * base * height. However, we don't know the height of the triangle and the given information doesn't directly tell us the height either.

In this case, we can use trigonometry to find the height. Let's consider the angle of 15 degrees. The side opposite to this angle is the height of the triangle we are looking for.

Since we know the length of the side between the 15 and 120 angles is 8 units, we can use the trigonometric function tangent to find the height. The tangent of an angle is defined as the ratio of the length of the opposite side to the length of the adjacent side.

So, we have the equation tan(15 degrees) = height / 8. By rearranging the equation, we can solve for the height. We find height = 8 * tan(15 degrees).

Now that we have the height, we can substitute the values into the formula for the area of a triangle to calculate the area.

A = (1/2) * base * height = (1/2) * 8 * tan(15 degrees).

Calculating this expression will give you the area of the triangle.

Let A = 15, B = 120 and C = 45 degrees

The side with length 8 is opposite the 45 degree angle. (side c)
sinC/c = sinB/b = sin A/a = 0.08839
b = 9.798
a = 2.928
c = 8.000

Area = (1/2) bc sin A
= 10.14 square length units