Questions

The user has provided an image of a right-angled triangle with two sides given and asks to find the third side.

From the image, we have a right-angled triangle with:

• One leg (height) = 3
• The hypotenuse = 4
• The other leg (base) is unknown, let's call it $x$.

We can use the Pythagorean theorem, which states that in a right-angled 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 (legs).

The Pythagorean theorem is: $a^2 + b^2 = c^2$ where $a$ and $b$ are the lengths of the legs, and $c$ is the length of the hypotenuse.

In this case: $a = x$ $b = 3$ $c = 4$

Substitute these values into the theorem: $x^2 + 3^2 = 4^2$ $x^2 + 9 = 16$

Now, solve for $x^2$: $x^2 = 16 - 9$ $x^2 = 7$

Now, solve for $x$ by taking the square root of both sides: $x = \sqrt{7}$

This is the exact answer.

To find the approximation, we calculate the numerical value of $\sqrt{7}$: $\sqrt{7} \approx 2.645751311$

Rounding to the nearest thousandth (three decimal places) as requested: $x \approx 2.646$

Exact Answer: $\sqrt{7}$ Approximation: $2.646$