Question

In ΔFGH, f = 4.8 cm, g = 5.2 cm and h=3.2 cm. Find the measure of ∠H to the nearest 10th of a degree.
Answer
Attempt 1 out of 2

Answer 1

To find the measure of angle ∠H in triangle ΔFGH with sides f = 4.8 cm, g = 5.2 cm, and h = 3.2 cm, we can use the Law of Cosines.

The Law of Cosines states that for any triangle with sides a, b, and c opposite to angles A, B, and C respectively:

\[
c^2 = a^2 + b^2 - 2ab \cdot \cos(C)
\]

In this case, we want to find angle ∠H. We can designate:
- \( f = 4.8 \) cm (side opposite ∠H)
- \( g = 5.2 \) cm (side opposite ∠F)
- \( h = 3.2 \) cm (side opposite ∠G)

Using the Law of Cosines to find ∠H:

\[
h^2 = f^2 + g^2 - 2fg \cdot \cos(H)
\]

Plugging in the values:

\[
(3.2)^2 = (4.8)^2 + (5.2)^2 - 2 \cdot (4.8) \cdot (5.2) \cdot \cos(H)
\]

Calculating the squares:

\[
10.24 = 23.04 + 27.04 - 2 \cdot (4.8) \cdot (5.2) \cdot \cos(H)
\]

Combine the squares:

\[
10.24 = 50.08 - 2 \cdot (4.8) \cdot (5.2) \cdot \cos(H)
\]

Now, calculate \( 2 \cdot (4.8) \cdot (5.2) \):

\[
2 \cdot (4.8) \cdot (5.2) = 49.92
\]

Substituting this in gives:

\[
10.24 = 50.08 - 49.92 \cdot \cos(H)
\]

Rearranging the equation:

\[
49.92 \cdot \cos(H) = 50.08 - 10.24
\]
\[
49.92 \cdot \cos(H) = 39.84
\]
\[
\cos(H) = \frac{39.84}{49.92}
\]

Calculating the cosine:

\[
\cos(H) \approx 0.797
\]

Now, we find the angle:

\[
H = \cos^{-1}(0.797)
\]

Calculating this using a calculator:

\[
H \approx 37.2^\circ
\]

Therefore, the measure of ∠H to the nearest tenth of a degree is:

\[
\boxed{37.2^\circ}
\]