MATH 1234 : algebra 2 - Gatesville H S - Free Printable

Here are the step-by-step solutions for the problems shown in the worksheet.

Part 1: Determine the missing value "x" (Problems 1–6)

These problems use the Pythagorean Theorem: $a^2 + b^2 = c^2$, where $c$ is the hypotenuse (the longest side opposite the right angle).

1.
* Legs are 9 and 12. Hypotenuse is $x$.
* $9^2 + 12^2 = x^2$
* $81 + 144 = x^2$
* $225 = x^2$
* $\sqrt{225} = 15$
* $x = 15$

2.
* Leg is 5, Hypotenuse is 13. Missing leg is $x$.
* $x^2 + 5^2 = 13^2$
* $x^2 + 25 = 169$
* $x^2 = 169 - 25$
* $x^2 = 144$
* $\sqrt{144} = 12$
* $x = 12$

3.
* Legs are 7 and 24. Hypotenuse is $x$.
* $7^2 + 24^2 = x^2$
* $49 + 576 = x^2$
* $625 = x^2$
* $\sqrt{625} = 25$
* $x = 25$

4.
* Leg is 15, Hypotenuse is 17. Missing leg is $x$.
* $x^2 + 15^2 = 17^2$
* $x^2 + 225 = 289$
* $x^2 = 289 - 225$
* $x^2 = 64$
* $\sqrt{64} = 8$
* $x = 8$

5.
* Legs are 8 and 15. Hypotenuse is $x$.
* $8^2 + 15^2 = x^2$
* $64 + 225 = x^2$
* $289 = x^2$
* $\sqrt{289} = 17$
* $x = 17$

6.
* Leg is 12, Hypotenuse is 20. Missing leg is $x$.
* $x^2 + 12^2 = 20^2$
* $x^2 + 144 = 400$
* $x^2 = 400 - 144$
* $x^2 = 256$
* $\sqrt{256} = 16$
* $x = 16$

---

Part 2: Use the distance formula to find the length of each line segment (Problems 7–9)

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

7. Points: $P(1, 1)$ and $Q(4, 5)$
* Change in $x$: $4 - 1 = 3$
* Change in $y$: $5 - 1 = 4$
* Distance squared: $3^2 + 4^2 = 9 + 16 = 25$
* Distance: $\sqrt{25} = 5$
* Length = 5

8. Points: $R(-2, 3)$ and $S(4, -5)$
* Change in $x$: $4 - (-2) = 6$
* Change in $y$: $-5 - 3 = -8$
* Distance squared: $6^2 + (-8)^2 = 36 + 64 = 100$
* Distance: $\sqrt{100} = 10$
* Length = 10

9. Points: $T(-6, 2)$ and $U(6, 7)$
* Change in $x$: $6 - (-6) = 12$
* Change in $y$: $7 - 2 = 5$
* Distance squared: $12^2 + 5^2 = 144 + 25 = 169$
* Distance: $\sqrt{169} = 13$
* Length = 13

Final Answer:
1. x = 15
2. x = 12
3. x = 25
4. x = 8
5. x = 17
6. x = 16
7. Length = 5
8. Length = 10
9. Length = 13