Two Sides And An Angle Are Given Below. Determine Whether The Given Information Results In One Triangle, Two Triangles, Or No Triangle At All. Solve Any Resulting Triangles.Given: $a = 4, B = 10, A = 70^{\circ}$Select The Correct Choice Below

Introduction

In geometry, a triangle is a polygon with three sides and three angles. When given two sides and an angle, we can determine whether the given information results in one triangle, two triangles, or no triangle at all. In this article, we will explore the process of determining the number of triangles given two sides and an angle, and solve any resulting triangles.

The Law of Sines

The Law of Sines is a fundamental concept in trigonometry that relates the lengths of the sides of a triangle to the sines of its angles. The Law of Sines states that for any triangle with sides of length a, b, and c, and angles A, B, and C, respectively:

$$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$

Applying the Law of Sines

Given two sides and an angle, we can use the Law of Sines to determine whether the given information results in one triangle, two triangles, or no triangle at all. Let's consider the given information:

• $a = 4$
• $b = 10$
• $A = 70^{\circ}$

We can use the Law of Sines to find the measure of angle B:

$$\frac{a}{\sin A} = \frac{b}{\sin B}$$

$$\frac{4}{\sin 70^{\circ}} = \frac{10}{\sin B}$$

$$\sin B = \frac{10 \sin 70^{\circ}}{4}$$

$$\sin B = 2.5 \sin 70^{\circ}$$

Using a calculator, we find that:

$$\sin B \approx 2.5 \times 0.9397$$

$$\sin B \approx 2.34925$$

Since the sine of an angle cannot be greater than 1, we have a problem. The given information does not result in a valid triangle.

Conclusion

In this article, we have explored the process of determining the number of triangles given two sides and an angle. We have used the Law of Sines to find the measure of angle B, and found that the given information does not result in a valid triangle. Therefore, the correct answer is:

• No triangle at all.

Solving the Resulting Triangles

Although the given information does not result in a valid triangle, we can still solve the resulting triangles. Let's consider the two possible triangles:

• Triangle 1: $a = 4, b = 10, A = 70^{\circ}$
• Triangle 2: $a = 4, b = 10, A = 110^{\circ}$

We can use the Law of Sines to find the measure of angle C for each triangle:

• Triangle 1:

$$\frac{a}{\sin A} = \frac{c}{\sin C}$$

$$\frac{4}{\sin 70^{\circ}} = \frac{c}{\sin C}$$

$$c = \frac{4 \sin C}{\sin 70^{\circ}}$$

Using a calculator, we find that:

$$c \approx \frac{4 \sin C}{0.9397}$$

$$c \approx 4.27 \sin C$$

Since the length of side c cannot be greater than the sum of the lengths of sides a and b, we have a problem. The given information does not result in a valid triangle.

• Triangle 2:

$$\frac{a}{\sin A} = \frac{c}{\sin C}$$

$$\frac{4}{\sin 110^{\circ}} = \frac{c}{\sin C}$$

$$c = \frac{4 \sin C}{\sin 110^{\circ}}$$

Using a calculator, we find that:

$$c \approx \frac{4 \sin C}{0.9397}$$

$$c \approx 4.27 \sin C$$

Since the length of side c cannot be greater than the sum of the lengths of sides a and b, we have a problem. The given information does not result in a valid triangle.

Conclusion

In this article, we have explored the process of determining the number of triangles given two sides and an angle. We have used the Law of Sines to find the measure of angle B, and found that the given information does not result in a valid triangle. Therefore, the correct answer is:

• No triangle at all.

Final Answer

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Introduction

In our previous article, we explored the process of determining the number of triangles given two sides and an angle. We used the Law of Sines to find the measure of angle B, and found that the given information does not result in a valid triangle. In this article, we will answer some frequently asked questions related to determining the number of triangles given two sides and an angle.

Q: What is the Law of Sines?

A: The Law of Sines is a fundamental concept in trigonometry that relates the lengths of the sides of a triangle to the sines of its angles. The Law of Sines states that for any triangle with sides of length a, b, and c, and angles A, B, and C, respectively:

$$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$

Q: How do I use the Law of Sines to determine the number of triangles given two sides and an angle?

A: To use the Law of Sines to determine the number of triangles given two sides and an angle, follow these steps:

1. Write down the given information, including the lengths of the two sides and the measure of the given angle.
2. Use the Law of Sines to find the measure of the other angle.
3. Check if the measure of the other angle is valid (i.e., between 0° and 180°).
4. If the measure of the other angle is valid, use the Law of Sines to find the length of the third side.
5. Check if the length of the third side is valid (i.e., less than the sum of the lengths of the other two sides).
6. If the length of the third side is valid, then the given information results in a valid triangle.

Q: What if the measure of the other angle is not valid?

A: If the measure of the other angle is not valid, then the given information does not result in a valid triangle. This can happen if the given angle is obtuse (greater than 90°) or if the given information is inconsistent.

Q: What if the length of the third side is not valid?

A: If the length of the third side is not valid, then the given information does not result in a valid triangle. This can happen if the given information is inconsistent or if the given angle is obtuse.

Q: Can I use the Law of Sines to find the measure of an angle given two sides and the measure of another angle?

A: Yes, you can use the Law of Sines to find the measure of an angle given two sides and the measure of another angle. To do this, follow these steps:

1. Write down the given information, including the lengths of the two sides and the measure of the given angle.
2. Use the Law of Sines to find the measure of the other angle.
3. Check if the measure of the other angle is valid (i.e., between 0° and 180°).

Q: Can I use the Law of Sines to find the length of a side given two angles and the length of another side?

A: Yes, you can use the Law of Sines to find the length of a side given two angles and the length of another side. To do this, follow these steps:

1. Write down the given information, including the lengths of the two sides and the measures of the two angles.
2. Use the Law of Sines to find the length of the third side.
3. Check if the length of the third side is valid (i.e., less than the sum of the lengths of the other two sides).

Conclusion

In this article, we have answered some frequently asked questions related to determining the number of triangles given two sides and an angle. We have used the Law of Sines to find the measure of an angle given two sides and the measure of another angle, and to find the length of a side given two angles and the length of another side. We have also discussed what to do if the measure of the other angle is not valid or if the length of the third side is not valid.

Final Answer

The final answer is: $\boxed{0}$