Let's analyze and solve the two problems provided in the image:

Problem 7: Ramp and Truck

We are tasked with finding the horizontal distance from where the ramp reaches the ground to the truck, given:

• The ramp is 12 feet long.
• The height of the truck bed is 3.5 feet.

Solution:

This problem involves a right triangle, where:

• The ramp is the hypotenuse ($c$).
• The height of the truck bed is one leg ($a$).
• The horizontal distance is the other leg ($b$).

Using the Pythagorean Theorem: $a^2 + b^2 = c^2$

Substitute the given values: $3.5^2 + b^2 = 12^2$

Calculate: $12.25 + b^2 = 144$

$b^2 = 144 - 12.25 = 131.75$

$b = \sqrt{131.75} \approx 11.48 \, \text{feet}$

Answer: The horizontal distance is approximately 11.48 feet.

Problem 8: Square and Diagonal

We are tasked with finding the perimeter of a square, given that its diagonal is 11.3 meters.

Solution:

For a square:

• The diagonal ($d$) is related to the side length ($s$) by: $d = s \sqrt{2}$

Rearrange to solve for $s$: $s = \frac{d}{\sqrt{2}}$

Substitute the given diagonal: $s = \frac{11.3}{\sqrt{2}} \approx \frac{11.3}{1.414} \approx 7.99 \, \text{meters}$

The perimeter ($P$) of a square is: $P = 4s$

Substitute the side length: $P = 4 \times 7.99 \approx 31.96 \, \text{meters}$

Answer: The perimeter is approximately 31.96 meters.

Do you want further clarification or additional steps?

Here are some related questions to expand your understanding:

1. How can we derive the Pythagorean Theorem from geometric principles?
2. What would the horizontal distance be if the ramp were shorter, e.g., 10 feet?
3. What is the area of the square in Problem 8, given the side length is approximately 7.99 meters?
4. How does the diagonal of a square scale with its side length?
5. Can the perimeter of a square be estimated directly from its diagonal using a simplified formula?

Tip: In problems involving right triangles, always ensure the hypotenuse is the longest side before applying the Pythagorean Theorem.