Let's solve the problems step by step.

1. Length of the diagonal of the swimming pool

• Given:
  • Length = 50 feet
  • Width = 20 feet
• Use the Pythagorean theorem: $c = \sqrt{a^2 + b^2}$ $c = \sqrt{50^2 + 20^2} = \sqrt{2500 + 400} = \sqrt{2900} \approx 53.9$ Answer: 53.9 feet

2. Distance between Cincinnati and San Diego

• Cincinnati at $(5, 4)$, San Diego at $(-10, -3)$
• Use the distance formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ $d = \sqrt{(-10 - 5)^2 + (-3 - 4)^2} = \sqrt{(-15)^2 + (-7)^2} = \sqrt{225 + 49} = \sqrt{274} \approx 16.6$ Answer: 16.6 units

3. Check if distances form a right triangle

• Given:
  • Katie to Ralph = 22.5 feet
  • Ralph to Juan = 58.5 feet
  • Juan to Katie = 54 feet
• Check if the Pythagorean theorem holds: $22.5^2 + 54^2 \stackrel{?}{=} 58.5^2$ $506.25 + 2916 = 3422.25 \quad \text{and} \quad 58.5^2 = 3422.25$ Yes, they form a right triangle.

4. Length of the diagonal of the picture frame

• Given:
  • Length = 7 inches
  • Width = 5 inches
• Use the Pythagorean theorem: $c = \sqrt{a^2 + b^2}$ $c = \sqrt{7^2 + 5^2} = \sqrt{49 + 25} = \sqrt{74} \approx 8.6$ Answer: 8.6 inches

5-10. Find the distance between points

• Use the distance formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

(5) $(0, 5)$ and $(-4, 2)$

$d = \sqrt{(-4 - 0)^2 + (2 - 5)^2} = \sqrt{(-4)^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5$ Answer: 5

(6) $(1, 9)$ and $(6, 3)$

$d = \sqrt{(6 - 1)^2 + (3 - 9)^2} = \sqrt{(5)^2 + (-6)^2} = \sqrt{25 + 36} = \sqrt{61} \approx 7.8$ Answer: 7.8

(7) $(-6, 4)$ and $(2, -6)$

$d = \sqrt{(2 - (-6))^2 + (-6 - 4)^2} = \sqrt{(8)^2 + (-10)^2} = \sqrt{64 + 100} = \sqrt{164} \approx 12.8$ Answer: 12.8

(8) $(-1, -7)$ and $(-3, -5)$

$d = \sqrt{(-3 - (-1))^2 + (-5 - (-7))^2} = \sqrt{(-2)^2 + (2)^2} = \sqrt{4 + 4} = \sqrt{8} \approx 2.8$ Answer: 2.8

(9) $(4, 0)$ and $(-9, 7)$

$d = \sqrt{(-9 - 4)^2 + (7 - 0)^2} = \sqrt{(-13)^2 + (7)^2} = \sqrt{169 + 49} = \sqrt{218} \approx 14.8$ Answer: 14.8

(10) $(0, -8)$ and $(4, 0)$

$d = \sqrt{(4 - 0)^2 + (0 - (-8))^2} = \sqrt{(4)^2 + (8)^2} = \sqrt{16 + 64} = \sqrt{80} \approx 8.9$ Answer: 8.9

11-16. Check if side lengths form a right triangle

• Use the Pythagorean theorem: $a^2 + b^2 = c^2$

(11) $7, 24, 25$

$7^2 + 24^2 = 49 + 576 = 625, \quad 25^2 = 625$ Yes

(12) $30, 40, 45$

$30^2 + 40^2 = 900 + 1600 = 2500, \quad 45^2 = 2025$ No

(13) $21.6, 28.8, 36$

$21.6^2 + 28.8^2 = 466.56 + 829.44 = 1296, \quad 36^2 = 1296$ Yes

(14) $10, 15, 18$

$10^2 + 15^2 = 100 + 225 = 325, \quad 18^2 = 324$ No

(15) $10.5, 36, 50$

$10.5^2 + 36^2 = 110.25 + 1296 = 1406.25, \quad 50^2 = 2500$ No

(16) $2.5, 6, 6.5$

$2.5^2 + 6^2 = 6.25 + 36 = 42.25, \quad 6.5^2 = 42.25$ Yes

Let me know if you need further clarification or details!

5 Related Questions:

1. What are the steps for verifying a right triangle with side lengths?
2. How is the distance formula derived from the Pythagorean theorem?
3. Why is rounding to the nearest tenth important in real-life applications?
4. What happens if the calculated hypotenuse is slightly off due to rounding?
5. Can we apply these formulas in three-dimensional space? How?

Tip: Always double-check your work when squaring and adding large numbers to avoid errors.