Question

Solve the following triangle for all missing information.

Given for Δ MVH: m = 15.7 cm, angle M = 43° and v = 18.5 cm. Show your work and include a sketch. If more than one answer exists, sketch both triangles with all measurements included.

Answer 1

To solve the triangle, we can use the Law of Sines and Law of Cosines.

First, let's draw a sketch of the triangle. Label the sides as a, b, and c, and the angles as A, B, and C.

c
M- *--------
| \
a | \ b
| \
H- *----------
v

The given information is:
m = 15.7 cm
Angle M = 43°
v = 18.5 cm

1. Using the Law of Sines:
We can use the Law of Sines to find angle H and side h.
The Law of Sines states:
a/sin A = b/sin B = c/sin C

sin H / m = sin M / v
sin H = (m * sin M) / v
sin H = (15.7 cm * sin 43°) / 18.5 cm
sin H ≈ 0.7955
H ≈ sin^(-1)(0.7955) ≈ 52.9°

Now, we can find h using the Law of Sines:
h / sin H = v / sin M
h = (v * sin H) / sin M
h = (18.5 cm * sin 52.9°) / sin 43°
h ≈ 21.65 cm

2. Using the Law of Cosines:
We can use the Law of Cosines to find side h:
h^2 = m^2 + v^2 - 2mv*cos M
h^2 = (15.7 cm)^2 + (18.5 cm)^2 - 2 * 15.7 cm * 18.5 cm * cos 43°
h^2 ≈ 327.94 cm^2
h ≈ √327.94 cm ≈ 18.11 cm

3. Sketching both triangles:
Depending on whether we use the Law of Sines or the Law of Cosines, we might get two different solutions for side h.

Triangle 1:
- Using the Law of Sines: h ≈ 21.65 cm
- Using the Law of Cosines: h ≈ 18.11 cm

Triangle 2:
- Using the Law of Sines: h ≈ 18.11 cm
- Using the Law of Cosines: h ≈ 21.65 cm

Thus, there are two possible triangles that satisfy the given information.