A segment that is a units long makes a C-degree angle with a segment that is b

units long. In terms of a, b, and C, find the third side of the triangle defined by this SAS description. You have done numerical versions of this question. Start by finding the length
of the altitude drawn to side b, as well as the length of the perpendicular projection of side
a onto side b. The resulting formula is known as the Law of Cosines.

3. What is the area of the triangle defined by a, b, and C?

To find the length of the third side of the triangle, we'll use the Law of Cosines. The Law of Cosines states that in a triangle with sides of lengths a, b, and c, and an angle C opposite the side c, we have the following formula:

c^2 = a^2 + b^2 - 2ab * cos(C)

In this case, we are given the lengths of sides a and b, and the angle C between them.

1. First, find the length of the altitude drawn to side b:
- The altitude drawn to side b will form a right triangle with the base b and the altitude as the hypotenuse.
- The length of the altitude can be found using the formula for the area of a triangle: Area = (1/2) * base * height.
- Rearrange the formula to find the height: height = (2 * Area) / base.
- Substitute the given values for a, b, and C, and calculate the height.

2. Next, find the length of the perpendicular projection of side a onto side b:
- The perpendicular projection of side a onto side b is the length of the segment that is perpendicular to side b and connects the intersection point of the altitude and side b to side a.
- This length is equal to the height of the altitude.

3. Now, use the Law of Cosines to find the length of the third side, c:
- Substitute the given values for a, b, and C into the Law of Cosines formula.
- Simplify the equation and solve for c.

To find the area of the triangle defined by a, b, and C, you can use the formula for the area of a triangle:

Area = (1/2) * base * height

Substitute the values for a, b, and the height (calculated earlier), and calculate the area.

To find the length of the altitude drawn to side b in the triangle defined by the SAS description, we can use the Law of Cosines.

The Law of Cosines states that in a triangle with sides a, b, and c and an angle C opposite side c, the following equation holds:

c^2 = a^2 + b^2 - 2ab*cos(C)

In this case, we want to find the length of the altitude drawn to side b, which would be the length of the perpendicular projection of side a onto side b. Let's call this length h.

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

Area = (1/2) * base * height

In this case, the base is side b and the height is h. Rearranging the formula gives:

h = 2 * (Area / b)

So, to find the length of the altitude drawn to side b, we need to know the area of the triangle.

Now let's move on to finding the area of the triangle defined by a, b, and C.

To find the area of a triangle, we can use Heron's formula. Heron's formula states that in a triangle with sides a, b, and c, the area can be calculated as:

Area = sqrt(s * (s - a) * (s - b) * (s - c))

where s is the semi-perimeter of the triangle, calculated as:

s = (a + b + c) / 2

In our case, the sides of the triangle are a, b, and c (not given). The angles of the triangle are A, B, and C, respectively. Note that if we know two sides and the angle between them (SAS), we can use the Law of Cosines to find the length of the third side.

So, unfortunately, without knowing the lengths of the sides a, b, and c or any additional information, we cannot directly find the area of the triangle or calculate the length of the altitude drawn to side b.

find the third side with the law of cosines.

Then, area= sqrt((s-a)(s-b)(s-c)s)

where s is 1/2 the perimeter. Perimeter then is a+b+thirdside.