Find the area of the triangle. Round to the nearest tenth.
B= 42.8° a= 12.7 c= 5.8
I found b by using the Pythagorean theorem and got
b= 11.298
but I don't know what to do next after this. I would find the altitude but I don't know angles A and C?
In a general triangle ABC
Area = (1/2)(ac)sin B
= (1/2)(12.7)(5.8)sin42.8º
= ....
You can only use the Pythagorean theorom with right triangles. This is not one.
The length of side b is given by the law of cosines
b^2 = a^2 + c^2 - 2 ac cos B
= 161.29 + 33.64 - 108.09 = 86.84
b = 9.319
Then use the law of sines to get angles A or C.
b/sin B = a/sin A = c/sin C = 13.716
sin A = a/13.716 = 0.926
A = 112.2 degrees (obtuse)
sin C = 0.423
C = 25.0 degrees
With all that information, you can get the area various ways.
To find the area of the triangle, you can use the formula for the area of a triangle:
Area = (1/2) * base * height
In this case, you have the lengths of sides a, b, and c. To find the height, you can use the formula for the area of a triangle:
Area = (1/2) * base * height
Since side b is the base and you have already calculated it as 11.298, you need to find the height. Let's call it h.
First, you can use the Law of Sines to find one of the angles (either angle A or C). The Law of Sines states that for any triangle:
sin(A)/a = sin(B)/b = sin(C)/c
In this case, you can use angle B and side b:
sin(B)/b = sin(A)/a
Plugging in the values:
sin(42.8°)/11.298 = sin(A)/12.7
Now, you can solve for sin(A):
sin(A) = (sin(42.8°)/11.298) * 12.7
Using a calculator, you can find:
sin(A) ≈ 0.483
To find angle A, you can take the inverse sine (sin⁻¹) of 0.483:
A ≈ sin⁻¹(0.483)
A ≈ 28.9°
Now that you know angle A, you can find angle C by using the fact that the sum of angles in a triangle is 180°:
B + A + C = 180°
42.8° + 28.9° + C = 180°
C ≈ 108.3°
With angles A and C known, you can find the height h by using trigonometry. In this case, you can use the sine function:
sin(C) = h/b
Plugging in the values:
sin(108.3°) = h/11.298
Now, solve for h:
h ≈ 11.298 * sin(108.3°)
Using a calculator, you can find:
h ≈ 11.298 * 0.933
h ≈ 10.539
Finally, you can calculate the area of the triangle using the formula:
Area = (1/2) * base * height
Area ≈ (1/2) * 11.298 * 10.539
Using a calculator, you can find:
Area ≈ 59.4
Therefore, the area of the triangle is approximately 59.4 square units (rounded to the nearest tenth).
To find the area of the triangle, you can use the formula:
Area = (1/2) * base * height
The base of the triangle is the side c, and the height can be found by drawing an altitude from vertex B to side c.
To find the height, we can use the sine function, since we know the angle opposite the side c and the length of side b.
Step 1: Find angle A:
Since we know angle B and angle C, we can find angle A by using the fact that the sum of all angles in a triangle is 180 degrees.
Angle A = 180 - (angle B + angle C)
Step 2: Find the height:
Using the sine function, we have:
sin(A) = height / side b
height = side b * sin(A)
Step 3: Calculate the area:
Area = (1/2) * base * height
Area = (1/2) * side c * height
Putting it all together:
1. Calculate angle A:
Angle A = 180 - (42.8 + angle C)
2. Find the height:
height = side b * sin(A)
3. Calculate the area:
Area = (1/2) * side c * height
Now, let's plug in the values you provided:
Angle A = 180 - (42.8 + angle C)
Angle A = 180 - (42.8 + angle C)
Angle A ≈ 180 - (42.8 + 90.2) [assuming angle C is a right angle]
Angle A ≈ 180 - 133
Angle A ≈ 47 degrees
height = side b * sin(A)
height = 11.298 * sin(47)
height ≈ 8.345
Area = (1/2) * side c * height
Area = (1/2) * 5.8 * 8.345
Area ≈ 23.8 square units (rounded to the nearest tenth)