Let's solve each problem step by step.

Directions: Find the value of $x$.

1. Triangle with sides 10, 7, and $x$. (Right triangle)

Using the Pythagorean theorem: $a^2 + b^2 = c^2$ Here, $c = 10$ (hypotenuse), $a = 7$, and $b = x$. Substituting: $7^2 + x^2 = 10^2$ $49 + x^2 = 100$ $x^2 = 100 - 49 = 51$ $x = \sqrt{51} \approx 7.14$

2. Triangle with sides $x$, 19, and 21. (Right triangle)

Here, $c = 21$ (hypotenuse), $a = 19$, and $b = x$. Substituting into the Pythagorean theorem: $19^2 + x^2 = 21^2$ $361 + x^2 = 441$ $x^2 = 441 - 361 = 80$ $x = \sqrt{80} = \sqrt{16 \cdot 5} = 4\sqrt{5} \approx 8.94$

3. Triangle with sides 27, 16, and $x$. (Right triangle)

Here, $c = x$ (hypotenuse), $a = 27$, and $b = 16$. Substituting: $27^2 + 16^2 = x^2$ $729 + 256 = x^2$ $x^2 = 985$ $x = \sqrt{985} \approx 31.42$

5. Triangle with sides 18, $x$, and 20. (Right triangle)

Here, $c = 20$ (hypotenuse), $a = 18$, and $b = x$. Substituting: $18^2 + x^2 = 20^2$ $324 + x^2 = 400$ $x^2 = 400 - 324 = 76$ $x = \sqrt{76} = \sqrt{4 \cdot 19} = 2\sqrt{19} \approx 8.7178$

6. Isosceles trapezoid with bases 31, $x$, and legs 17 and 19.

The goal is to solve for $x$, the upper base. By splitting the trapezoid into two right triangles and a rectangle, the height of the triangles is given as 17. Let $d$ be the horizontal length of each triangle's base, so: $d^2 + 17^2 = 19^2$ $d^2 + 289 = 361$ $d^2 = 72 \quad \implies \quad d = \sqrt{72} = 6\sqrt{2} \approx 8.485$ The total base length is $31$, so: $x = 31 - 2d = 31 - 2(8.485) \approx 31 - 16.97 \approx 14.03$

8. Ramp problem.

The ramp length is 12 feet, the height from the ground is 3.5 feet, and the horizontal distance ($x$) is unknown. Using the Pythagorean theorem: $x^2 + 3.5^2 = 12^2$ $x^2 + 12.25 = 144$ $x^2 = 144 - 12.25 = 131.75$ $x = \sqrt{131.75} \approx 11.48 \, \text{feet.}$

Summary of Results:

1. $x \approx 7.14$
2. $x \approx 8.94$
3. $x \approx 31.42$
4. $x \approx 8.72$
5. $x \approx 14.03$
6. $x \approx 11.48 \, \text{feet}$

Would you like detailed explanations for any specific step or concept?

Related Questions:

1. How does the Pythagorean theorem apply to real-world scenarios like ramps or ladders?
2. Can you explain the steps for simplifying square roots, like $\sqrt{76}$ or $\sqrt{985}$?
3. Why is it important to check if the triangle is a right triangle before using the Pythagorean theorem?
4. How do you handle problems with trapezoids or other non-triangular shapes?
5. What are some practical ways to estimate square roots without a calculator?

Tip: Always double-check which side of the triangle is the hypotenuse before applying the Pythagorean theorem. This avoids common errors!