- GEOMEIRY Word problem involving the Pythagorean Theorem A $33-m$ tall building casts a shadow. The distance from the top of the building to the tip of the shadow is $37 \mathrm{~m}$. Find the length of the shadow. If necessary, round your answer to the nearest tenth.

Step 1 ：The problem is asking for the length of the shadow cast by the building. We know the height of the building and the distance from the top of the building to the tip of the shadow. This forms a right triangle, and we can use the Pythagorean theorem to solve for the length of the shadow. The Pythagorean theorem 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. In this case, the hypotenuse is the distance from the top of the building to the tip of the shadow, and one of the other sides is the height of the building. We can solve for the length of the shadow (the other side) using this theorem.

Step 2 ：Let's denote the height of the building as \(a\), the hypotenuse as \(c\), and the length of the shadow as \(b\). We know that \(a = 33m\) and \(c = 37m\).

Step 3 ：According to the Pythagorean theorem, we have \(c^2 = a^2 + b^2\). We can rearrange this equation to solve for \(b\): \(b = \sqrt{c^2 - a^2}\).

Step 4 ：Substituting the given values into the equation, we get \(b = \sqrt{37^2 - 33^2} = 16.73320053068151\).

Step 5 ：Rounding to the nearest tenth, we get \(b = 16.7m\).

Step 6 ：Final Answer: The length of the shadow is approximately \(\boxed{16.7}\) meters.