Question content area top left
Part 1
Determine the remaining sides and angles of the triangle ABC.
.
.
.
Question content area top right
Part 1
51 degrees51°
a equals 9 ma=9 m
54 degrees54°
bb
cc
Upper CC

A
B

To determine the remaining sides and angles of the triangle ABC, we need to use the given information and apply the appropriate trigonometric laws (such as the Law of Sines or the Law of Cosines).

Given:
- Angle A = 51°
- Side a = 9 m (opposite Angle A)
- Angle B = 54°

Since we have two angles, we can find the third angle using the fact that the sum of the angles in a triangle is 180°.

\[ \text{Angle C} = 180° - \text{Angle A} - \text{Angle B} \]
\[ \text{Angle C} = 180° - 51° - 54° \]
\[ \text{Angle C} = 75° \]

Now we know all three angles:
- Angle A = 51°
- Angle B = 54°
- Angle C = 75°

Next, we can use the Law of Sines to find the remaining sides \( b \) and \( c \).

The Law of Sines states:
\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]

First, solve for side \( b \):
\[ \frac{a}{\sin A} = \frac{b}{\sin B} \]
\[ \frac{9}{\sin 51°} = \frac{b}{\sin 54°} \]
\[ b = \frac{9 \cdot \sin 54°}{\sin 51°} \]

Using a calculator:
\[ \sin 51° \approx 0.7771 \]
\[ \sin 54° \approx 0.8090 \]

\[ b = \frac{9 \cdot 0.8090}{0.7771} \]
\[ b \approx \frac{7.281}{0.7771} \]
\[ b \approx 9.37 \, \text{m} \]

Now, solve for side \( c \):
\[ \frac{a}{\sin A} = \frac{c}{\sin C} \]
\[ \frac{9}{\sin 51°} = \frac{c}{\sin 75°} \]
\[ c = \frac{9 \cdot \sin 75°}{\sin 51°} \]

Using a calculator:
\[ \sin 75° \approx 0.9659 \]

\[ c = \frac{9 \cdot 0.9659}{0.7771} \]
\[ c \approx \frac{8.6931}{0.7771} \]
\[ c \approx 11.19 \, \text{m} \]

Therefore, the remaining sides and angles of the triangle ABC are:
- Angle A = 51°
- Angle B = 54°
- Angle C = 75°
- Side a = 9 m
- Side b ≈ 9.37 m
- Side c ≈ 11.19 m

Would you like more details or have any questions?

Here are 5 additional questions you might want to ask:
1. Can you explain the Law of Sines in more detail?
2. How do you find the sine of an angle using a calculator?
3. What is the Law of Cosines and when should it be used?
4. How do you determine the area of a triangle using trigonometry?
5. What are some common mistakes to avoid when solving triangles?

**Tip:** When solving trigonometric problems, always double-check angle measures and calculations to ensure accuracy.

Explore how to determine the remaining sides and angles of triangle ABC given Angle A = 51°, Side a = 9 m, and Angle B = 54°. This detailed solution uses the Law of Sines and angle properties, suitable for Grades 9-12 students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation