Find the exact value of each variable that represents a side length in a right triangle.

5 right triangles. Top left, Side lengths are: h, 8, 10. Top center, Side lengths are: 6, k, 6 point 5. Top right, side lengths are: 2, m, 5. Bottom left, Side lengths are: square root 10, n, 10. Bottom right, Side lengths are: p, square root 68, square root 85.

In each part, \(a\) and \(b\) represent the length of a leg of a right triangle, and \(c\) represents the length of its hypotenuse. Find the unknown length, given the other two lengths.

1. \(a=3, b=1, c={?}\)
2. \(a={?}, b=2, c=\sqrt{29}\)
3. \(a=\sqrt5, b=\sqrt7, c={?}\)
4. \(a=\sqrt8, b={?}, c=\sqrt{13}\)
5. \(a={?}, b=\sqrt{13}, c=4\)

What is the exact length of each line segment? Explain or show your reasoning. (Each grid square represents 1 square unit.)

A line segment labeled “q” on a square grid. The line segment starts at an intersection point on the grid and slants upward and to the right to an end point that is 4 units to the right and 5 units up.

In 2015, there were roughly \(1 \times 10^6\) high school football players and \(2 \times 10^3\) professional football players in the United States. About how many times more high school football players are there? Explain how you know.

Evaluate:

1. \(\left(\frac{1}{2}\right)^3\)

2. \(\left(\frac{1}{2}\right)^{\text-3}\)

Here is a scatter plot of weight vs. age for different Dobermanns. The model represented by \(y = 2.45x + 1.22\) is graphed with the scatter plot. Here, \(x\) represents age in weeks, and \(y\) represents weight in pounds.

Scatter plot with line of best fit. Horizontal axis, age in weeks, scale 0 to 25, by 5’s. Vertical axis, weight in pounds, scale 0 to 80, by 20’s. Line of best fit, slope=2.45, y intercept=1.22. 

1. What is the slope and what does it mean in this situation?
2. Based on this model, how heavy would you expect a newborn Dobermann to be?