5. An interior designer is creating a triangular tile of unusual shape. The lengths of the til’s sides are 5.4cm, 7.0cm and 9.2cm. The designer wants to know the angle measurements so that he can plan designs using tiles of this shape. What are the angles of the designer’s new tile?
To find the angles of a triangle when side lengths are given, we can use the Law of Cosines, which states:
c^2 = a^2 + b^2 - 2ab * cos(C)
In this case, we have sides a = 5.4 cm, b = 7.0 cm, and c = 9.2 cm. We'll calculate each angle one at a time.
First, let's find angle C.
Plugging the given values into the Law of Cosines equation, we have:
9.2^2 = 5.4^2 + 7.0^2 - 2 * 5.4 * 7.0 * cos(C)
84.64 = 29.16 + 49 - 75.6 * cos(C)
84.64 = 78.16 - 75.6 * cos(C)
Now, let's isolate cos(C) by subtracting 78.16 from both sides:
84.64 - 78.16 = 78.16 - 75.6 * cos(C) - 78.16
6.48 = -75.6 * cos(C)
Divide both sides by -75.6:
6.48 / -75.6 = cos(C)
cos(C) ≈ -0.0857
Now, use the inverse cosine function (cos^(-1)) to find the measure of C in radians:
C ≈ cos^(-1)(-0.0857)
C ≈ 1.655 radians
To convert radians to degrees, multiply by 180/π:
C ≈ 1.655 * (180/π) ≈ 94.8 degrees
So, angle C is approximately 94.8 degrees.
To find angle A, we can use the Law of Sines, which states:
sin(A) / a = sin(C) / c
Plugging in the known values:
sin(A) / 5.4 = sin(94.8) / 9.2
Cross-multiplying:
sin(A) * 9.2 = 5.4 * sin(94.8)
Now, divide both sides by 9.2:
sin(A) = (5.4 * sin(94.8)) / 9.2
Now, use the inverse sine function (sin^(-1)) to find the measure of A in radians:
A ≈ sin^(-1)((5.4 * sin(94.8)) / 9.2)
A ≈ 0.673 radians
To convert radians to degrees, multiply by 180/π:
A ≈ 0.673 * (180/π) ≈ 38.6 degrees
So, angle A is approximately 38.6 degrees.
Finally, to find angle B, we can use the fact that the sum of angles in a triangle is always 180 degrees:
B = 180 - A - C
B ≈ 180 - 38.6 - 94.8
B ≈ 46.6 degrees
So, angle B is approximately 46.6 degrees.
Therefore, the angles of the designer's new tile are approximately:
A ≈ 38.6 degrees
B ≈ 46.6 degrees
C ≈ 94.8 degrees