The dimensions of a rectangle are given. For each rectangle, find the exact length of its diagonal (a segment whose endpoints lie on opposite vertices of the rectangle).

1. 10 by 24
2. 16 by 30
3. 7 by 7
4. \(\sqrt{13}\) by 6
5. \(\sqrt{3}\) by 1
6. \(\frac12\) by \(\frac{\sqrt{3}}{2}\)

A standard city block in Manhattan is a rectangle measuring 80 m by 270 m. A resident wants to get from one corner of a block to the opposite corner of a block that contains a park. She wonders about the difference in length between cutting across the diagonal through the park compared to going around the park, along the streets.

How much shorter would her walk be going through the park? Round your answer to the nearest meter.

Here is an equilateral triangle. The length of each side is 2 units. A height is drawn. In an equilateral triangle, the height divides the opposite side into two pieces of equal length.

1. Find the exact height.
2. Find the area of the equilateral triangle.
3. (Challenge) Using \(x\) for the length of each side in an equilateral triangle, express its area in terms of \(x\).

What is the side length of this square? Explain your reasoning.

Find all the solutions to each equation.

1. \(x^2=81\)
2. \(x^2=100\)
3. \(\sqrt{x}=12\)

What is the value of the "?" in the following equation? Explain or show your reasoning. 

\(10^3 \boldcdot 10^{?} = 10^8\)