In a right triangle, a and b are the lengths of the legs and c is the length of the hypotenuse. If b = 6 yards and c = 9 yards, what is a? If necessary, round to the nearest tenth.
Understand the Problem
The question provides the lengths of one leg (b) and the hypotenuse (c) of a right triangle and asks us to find the length of the other leg (a). We can use the Pythagorean theorem to solve for a.
Answer
$a \approx 6.7$ yards
Answer for screen readers
$a \approx 6.7$ yards
Steps to Solve
- State the Pythagorean theorem
The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b). This can be written as:
$$a^2 + b^2 = c^2$$
- Plug in the provided values
We are given that $b = 6$ and $c = 9$. Substituting these values into the Pythagorean theorem, we get:
$$a^2 + 6^2 = 9^2$$
- Simplify the equation
Squaring the known values:
$$a^2 + 36 = 81$$
- Isolate $a^2$
Subtract 36 from both sides of the equation:
$$a^2 = 81 - 36$$
$$a^2 = 45$$
- Solve for $a$
Take the square root of both sides of the equation:
$$a = \sqrt{45}$$
- Approximate the value of $a$ to the nearest tenth
$$\sqrt{45} \approx 6.7082$$
Rounding to the nearest tenth, we get $a \approx 6.7$.
$a \approx 6.7$ yards
More Information
The length of side $a$ is approximately 6.7 yards. This result is obtained by applying the Pythagorean theorem and rounding to the nearest tenth as requested.
Tips
A common mistake is to add $b^2$ to $c^2$ instead of subtracting it, leading to $a^2 = 81 + 36 = 117$, and $a = \sqrt{117} \approx 10.8$. Remember that the hypotenuse is always the longest side, so the legs must be shorter than the hypotenuse. Adding $b^2$ to $c^2$ will result in a value larger than the hypotenuse. Another common mistake can be not rounding to the nearest tenth at the end of the calculation.
AI-generated content may contain errors. Please verify critical information
