Question
how to find the length of a side in a triangle when two sides are given
Answers
GPT-4o mini
To find the length of a side in a triangle when two sides are given, you usually use the triangle inequalities or the Law of Cosines, depending on what information you have about the angles.
1. **Triangle Inequality Theorem**:
This theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. If you have two sides, \( a \) and \( b \), you can find the range for the third side \( c \) as follows:
- \( |a - b| < c < a + b \)
This gives you the possible lengths for side \( c \) but does not determine a unique length.
2. **Law of Cosines**:
If you know the lengths of two sides and the included angle (the angle between the two sides), you can find the length of the third side using the Law of Cosines:
\[
c^2 = a^2 + b^2 - 2ab \cdot \cos(C)
\]
where \( c \) is the length of the side opposite angle \( C \), and \( a \) and \( b \) are the lengths of the other two sides.
3. **Using Trigonometry**:
If you have an angle opposite one of the given sides (or the two sides and the included angle), you can use the Law of Sines as well:
\[
\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}
\]
Here, \( A \) and \( B \) are the angles opposite sides \( a \) and \( b \), respectively.
### Example:
Suppose you have a triangle with sides \( a = 7 \) and \( b = 10 \), and you know the angle \( C = 60^\circ \).
1. Use the Law of Cosines to find side \( c \):
\[
c^2 = 7^2 + 10^2 - 2 \cdot 7 \cdot 10 \cdot \cos(60^\circ)
\]
\[
c^2 = 49 + 100 - 70
\]
\[
c^2 = 79
\]
\[
c = \sqrt{79} \approx 8.89
\]
Alternatively, if you only knew \( a \) and \( b \), you would solely be able to determine the range of possible values for \( c \) using the triangle inequality.
Make sure you take care of the angle measures and the relationships between the sides when applying these rules.
1. **Triangle Inequality Theorem**:
This theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. If you have two sides, \( a \) and \( b \), you can find the range for the third side \( c \) as follows:
- \( |a - b| < c < a + b \)
This gives you the possible lengths for side \( c \) but does not determine a unique length.
2. **Law of Cosines**:
If you know the lengths of two sides and the included angle (the angle between the two sides), you can find the length of the third side using the Law of Cosines:
\[
c^2 = a^2 + b^2 - 2ab \cdot \cos(C)
\]
where \( c \) is the length of the side opposite angle \( C \), and \( a \) and \( b \) are the lengths of the other two sides.
3. **Using Trigonometry**:
If you have an angle opposite one of the given sides (or the two sides and the included angle), you can use the Law of Sines as well:
\[
\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}
\]
Here, \( A \) and \( B \) are the angles opposite sides \( a \) and \( b \), respectively.
### Example:
Suppose you have a triangle with sides \( a = 7 \) and \( b = 10 \), and you know the angle \( C = 60^\circ \).
1. Use the Law of Cosines to find side \( c \):
\[
c^2 = 7^2 + 10^2 - 2 \cdot 7 \cdot 10 \cdot \cos(60^\circ)
\]
\[
c^2 = 49 + 100 - 70
\]
\[
c^2 = 79
\]
\[
c = \sqrt{79} \approx 8.89
\]
Alternatively, if you only knew \( a \) and \( b \), you would solely be able to determine the range of possible values for \( c \) using the triangle inequality.
Make sure you take care of the angle measures and the relationships between the sides when applying these rules.