Let's work through each problem step-by-step.

Problem 10

We need to find a third number so that the three numbers form a right triangle. For any right triangle, the numbers should satisfy the Pythagorean theorem: $a^2 + b^2 = c^2$, where $c$ is the hypotenuse.

1. (i) Numbers 9 and 41

   • Let's assume 41 is the hypotenuse, so we have $9^2 + x^2 = 41^2$.
   • Solving: $9^2 + x^2 = 41^2$ $81 + x^2 = 1681$ $x^2 = 1600$ $x = \sqrt{1600} = 40$
   • Therefore, the third number is 40.

2. (ii) Numbers 13 and 85

   • Assuming 85 is the hypotenuse: $13^2 + x^2 = 85^2$ $169 + x^2 = 7225$ $x^2 = 7056$ $x = \sqrt{7056} = 84$
   • The third number is 84.

Problem 11

A right triangle has a hypotenuse of 13 and one leg of 5. We need to find the other leg.

• Using the Pythagorean theorem: $5^2 + x^2 = 13^2$ $25 + x^2 = 169$ $x^2 = 144$ $x = \sqrt{144} = 12$
• The other leg is 12.

Problem 12

Two joggers run 8 miles north and then 5 miles west. To find the shortest distance back, we use the Pythagorean theorem, where the legs are 8 and 5 miles.

• Calculate the hypotenuse: $8^2 + 5^2 = c^2$ $64 + 25 = c^2$ $c^2 = 89$ $c = \sqrt{89} \approx 9.4 \text{ miles}$
• The shortest distance is 9.4 miles.

Problem 13

Oscar's dog house has slanted sides of 5 feet, and the base is 6 feet across. This forms an isosceles triangle with a height we need to find.

1. The base is 6 feet, so each half of the base is 3 feet.
2. Using the Pythagorean theorem with one of the right triangles formed: $3^2 + h^2 = 5^2$ $9 + h^2 = 25$ $h^2 = 16$ $h = \sqrt{16} = 4$

• The height is 4 feet.

Problem 14

To avoid a pond, the path involves walking 34 meters south and 41 meters east. We need to find the direct distance.

• Using the Pythagorean theorem: $34^2 + 41^2 = c^2$ $1156 + 1681 = c^2$ $c^2 = 2837$ $c = \sqrt{2837} \approx 53.3 \text{ meters}$
• Distance saved by walking through the pond: $(34 + 41) - 53.3 = 75 - 53.3 = 21.7 \text{ meters}$
• 21.7 meters would be saved.

Problem 15

The suitcase measures 24 inches long with a diagonal of 30 inches. We need to find the width.

• Using the Pythagorean theorem: $24^2 + w^2 = 30^2$ $576 + w^2 = 900$ $w^2 = 324$ $w = \sqrt{324} = 18 \text{ inches}$
• The width is 18 inches.

Would you like further details on any of these solutions? Here are related questions to explore:

1. How can you determine if three numbers form a right triangle?
2. What is the significance of the Pythagorean theorem in geometry?
3. How does one calculate the distance saved in problem-solving involving direct and indirect paths?
4. What is the process of finding heights in isosceles triangles with given base and side lengths?
5. How can understanding triangles help in real-life applications?

Tip: When working with right triangles, always check if the side lengths satisfy $a^2 + b^2 = c^2$ for correct triangle configuration.