How do you find the length of an unknown leg in a night triangle?

To find the length of an unknown leg in a right triangle, you can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides (the legs). The formula is:

\[ c^2 = a^2 + b^2 \]

Where:

• \( c \) is the length of the hypotenuse,
• \( a \) and \( b \) are the lengths of the two legs.

If you know the lengths of one leg and the hypotenuse, you can rearrange the formula to find the unknown leg. For example, if you know one leg \( a \) and the hypotenuse \( c \), you can find the length of the other leg \( b \) by rearranging the formula:

\[ b = \sqrt{c^2 - a^2} \]

Similarly, if you know one leg \( b \) and the hypotenuse \( c \), you can find the length of the other leg \( a \):

\[ a = \sqrt{c^2 - b^2} \]

Make sure to follow these steps:

1. Identify the lengths you have (either a leg or the hypotenuse).
2. Substitute those values into the Pythagorean theorem.
3. Solve for the unknown leg by performing the necessary calculations.

If the triangle is not a right triangle but you know two sides and the included angle, you can use the Law of Cosines to find the unknown leg. The Law of Cosines states:

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

Where \( C \) is the included angle between the known legs \( a \) and \( b \), and \( c \) is the opposite side. Rearranging allows you to solve for any side as needed.

Feel free to provide specific values or conditions if you need help with a particular problem!

To find the length of an unknown leg in a right triangle, use the Pythagorean theorem, which states:

\[ c^2 = a^2 + b^2 \]

Where \( c \) is the hypotenuse and \( a \) and \( b \) are the legs. If you know one leg and the hypotenuse, rearrange the formula to find the unknown leg:

1. For leg \( b \): \[ b = \sqrt{c^2 - a^2} \]

2. For leg \( a \): \[ a = \sqrt{c^2 - b^2} \]

If the triangle is not a right triangle and you know two sides and the included angle, use the Law of Cosines:

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

This approach allows you to solve for any side of the triangle as needed.