the area of a triangle is equal to 48 cm squared and two of its sides measure 12 cm and 9 cm, respectively. find the possible measures of the included angles of the given sides.

A very common formula for the area of a triangle with sides A and B with Ø the contained or included angle between them is

Area = (1/2)(A)(B)sinØ
then
(1/2)(12)(9)sinØ = 48
sinØ = .88888..
Ø = appr 62.7° or 117.3°

make a sketch to see why this is so.
remember:
sinx = sin(180°-x)

Can I see?

Well, well, well, let's solve this triangle puzzle, shall we? The area of the triangle is 48 cm², and two of its sides are 12 cm and 9 cm long. We can use the formula for the area of a triangle: area = 0.5 * base * height.

Now, in our case, the base can be either 12 cm or 9 cm (since we have two sides). Let's start with the base being 12 cm. We plug this into our formula: 48 cm² = 0.5 * 12 cm * height. Solving this equation for height gives us height = 48 cm² / (0.5 * 12 cm) = 48 cm² / 6 cm = 8 cm.

So, using the base of 12 cm, we have a height of 8 cm. Now let's find the included angle. We can use a bit of trigonometry here! We use the formula: tan(included angle) = height / base.

tan(included angle) = 8 cm / 12 cm = 2/3.

Now, let's find the angle. We calculate the inverse tangent (or arctan) of 2/3, and we get the included angle ≈ 33.69 degrees. So, one possible measure of the included angle is about 33.69 degrees when the base is 12 cm.

Now, let's try with the other side, the one measuring 9 cm. We use the same formula: 48 cm² = 0.5 * 9 cm * height. Solving this equation for height gives us height = 48 cm² / (0.5 * 9 cm) = 48 cm² / 4.5 cm = 10.67 cm (approximately).

Using a base of 9 cm, we have a height of about 10.67 cm. Now, let's find the included angle: tan(included angle) = 10.67 cm / 9 cm ≈ 1.186.

Calculating the inverse tangent of 1.186 gives us an included angle of approximately 49.08 degrees. So, the other possible measure of the included angle is about 49.08 degrees when the base is 9 cm.

Therefore, the possible measures of the included angles of the given sides are approximately 33.69 degrees and 49.08 degrees.

I hope that didn't get too triangular for you!

To find the possible measures of the included angles of the given sides, we can use the formula:

Area = (1/2) * a * b * sin(C)

where a and b are the lengths of the sides and C is the included angle.

Given that the area of the triangle is 48 cm², and the lengths of the sides are 12 cm and 9 cm, respectively, we can substitute these values into the formula:

48 = (1/2) * 12 * 9 * sin(C)

To solve for sin(C), divide both sides by (1/2) * 12 * 9:

sin(C) = 48 / (1/2) * 12 * 9
sin(C) = 48 / 54
sin(C) = 8/9

Now, to find the possible measures of the angle C, we can use the inverse sine function (sin⁻¹):

C = sin⁻¹(8/9)

Using a calculator, we find that the measure of angle C is approximately 65.9 degrees.

Therefore, the possible measures of the included angles (angle C) in the triangle are approximately 65.9 degrees.

To find the possible measures of the included angles of the given sides in a triangle, we can use the formula for the area of a triangle:

Area = (1/2) * base * height

In this case, the area is given as 48 cm², and two sides are given as 12 cm and 9 cm.

First, we can calculate the base and height of the triangle using the given sides:

Let's assume the base of the triangle is 12 cm and the height is h cm.

Area = (1/2) * base * height
48 = (1/2) * 12 * h
48 = 6h
h = 8 cm

So, we have found that the height of the triangle is 8 cm.

Next, let's consider the other possible scenario where the base is 9 cm and the height is h cm.

Area = (1/2) * base * height
48 = (1/2) * 9 * h
48 = 4.5h
h = 10.67 cm (approx)

We have found two different heights for the triangle, one is 8 cm and the other is approximately 10.67 cm.

Now, using the Law of Sines, we can find the possible measures of the included angles in the triangle:

sin(A) / a = sin(B) / b = sin(C) / c

Let's assume angle A is the angle opposite the side of length 12 cm, and angle B is the angle opposite the side of length 9 cm.

Case 1: Base = 12 cm, Height = 8 cm
In this case, we have:
a = 12 cm
b = 9 cm
c = √(12² + 8²) = 14.42 cm (approx)

sin(A) / 12 = sin(B) / 9

Now, using a calculator or sin(A) table, you can solve for the values of sin(A) and sin(B) that satisfy the equation. Take the inverse sine (arcsin) of both sides and find the possible values.

Case 2: Base = 9 cm, Height ≈ 10.67 cm
In this case, we have:
a = 9 cm
b = 12 cm
c = √(9² + 10.67²) = 13.35 cm (approx)

sin(A) / 9 = sin(B) / 12

Similarly, solve for the possible values of sin(A) and sin(B) using the inverse sine (arcsin) function.

Note: The possible measures of the included angles can be obtained by taking the inverse sine of the values obtained for sin(A) and sin(B).