Sure! Let's solve each of the provided problems step-by-step using the Pythagorean Theorem, which states that [tex]\(a^2 + b^2 = c^2\)[/tex].
### 3. [tex]\(a = 3, c = 5\)[/tex]
To find [tex]\(b\)[/tex], we rearrange the Pythagorean Theorem:
[tex]\[b = \sqrt{c^2 - a^2}\][/tex]
Substituting the given values:
[tex]\[b = \sqrt{5^2 - 3^2}\][/tex]
[tex]\[b = \sqrt{25 - 9}\][/tex]
[tex]\[b = \sqrt{16}\][/tex]
[tex]\[b = 4.0\][/tex]
### 4. [tex]\(b = 3, c = 7\)[/tex]
To find [tex]\(a\)[/tex], we use:
[tex]\[a = \sqrt{c^2 - b^2}\][/tex]
Substituting the given values:
[tex]\[a = \sqrt{7^2 - 3^2}\][/tex]
[tex]\[a = \sqrt{49 - 9}\][/tex]
[tex]\[a = \sqrt{40}\][/tex]
[tex]\[a \approx 6.32\][/tex]
### 5. [tex]\(a = 5, b = 6\)[/tex]
To find [tex]\(c\)[/tex], we use:
[tex]\[c = \sqrt{a^2 + b^2}\][/tex]
Substituting the given values:
[tex]\[c = \sqrt{5^2 + 6^2}\][/tex]
[tex]\[c = \sqrt{25 + 36}\][/tex]
[tex]\[c = \sqrt{61}\][/tex]
[tex]\[c \approx 7.81\][/tex]
### 6. [tex]\(b = 5, c = 10\)[/tex]
To find [tex]\(a\)[/tex], we use:
[tex]\[a = \sqrt{c^2 - b^2}\][/tex]
Substituting the given values:
[tex]\[a = \sqrt{10^2 - 5^2}\][/tex]
[tex]\[a = \sqrt{100 - 25}\][/tex]
[tex]\[a = \sqrt{75}\][/tex]
[tex]\[a \approx 8.66\][/tex]
### 7. [tex]\(a = 8, b = 8\)[/tex]
To find [tex]\(c\)[/tex], we use:
[tex]\[c = \sqrt{a^2 + b^2}\][/tex]
Substituting the given values:
[tex]\[c = \sqrt{8^2 + 8^2}\][/tex]
[tex]\[c = \sqrt{64 + 64}\][/tex]
[tex]\[c = \sqrt{128}\][/tex]
[tex]\[c \approx 11.31\][/tex]
### 8. [tex]\(a = 5, b = 12\)[/tex]
To find [tex]\(c\)[/tex], we use:
[tex]\[c = \sqrt{a^2 + b^2}\][/tex]
Substituting the given values:
[tex]\[c = \sqrt{5^2 + 12^2}\][/tex]
[tex]\[c = \sqrt{25 + 144}\][/tex]
[tex]\[c = \sqrt{169}\][/tex]
[tex]\[c = 13.0\][/tex]
### 9. [tex]\(a = 8, b = 12\)[/tex]
To find [tex]\(c\)[/tex], we use:
[tex]\[c = \sqrt{a^2 + b^2}\][/tex]
Substituting the given values:
[tex]\[c = \sqrt{8^2 + 12^2}\][/tex]
[tex]\[c = \sqrt{64 + 144}\][/tex]
[tex]\[c = \sqrt{208}\][/tex]
[tex]\[c \approx 14.42\][/tex]
### 10. [tex]\(a = 7, c = 25\)[/tex]
To find [tex]\(b\)[/tex], we use:
[tex]\[b = \sqrt{c^2 - a^2}\][/tex]
Substituting the given values:
[tex]\[b = \sqrt{25^2 - 7^2}\][/tex]
[tex]\[b = \sqrt{625 - 49}\][/tex]
[tex]\[b = \sqrt{576}\][/tex]
[tex]\[b = 24.0\][/tex]
### 11. [tex]\(b = 15, c = 17\)[/tex]
To find [tex]\(a\)[/tex], we use:
[tex]\[a = \sqrt{c^2 - b^2}\][/tex]
Substituting the given values:
[tex]\[a = \sqrt{17^2 - 15^2}\][/tex]
[tex]\[a = \sqrt{289 - 225}\][/tex]
[tex]\[a = \sqrt{64}\][/tex]
[tex]\[a = 8.0\][/tex]
### 12. [tex]\(a = 9, c = 41\)[/tex]
To find [tex]\(b\)[/tex], we use:
[tex]\[b = \sqrt{c^2 - a^2}\][/tex]
Substituting the given values:
[tex]\[b = \sqrt{41^2 - 9^2}\][/tex]
[tex]\[b = \sqrt{1681 - 81}\][/tex]
[tex]\[b = \sqrt{1600}\][/tex]
[tex]\[b = 40.0\][/tex]
### 13. [tex]\(b = 3, c = 3.4\)[/tex]
To find [tex]\(a\)[/tex], we use:
[tex]\[a = \sqrt{c^2 - b^2}\][/tex]
Substituting the given values:
[tex]\[a = \sqrt{3.4^2 - 3^2}\][/tex]
[tex]\[a = \sqrt{11.56 - 9}\][/tex]
[tex]\[a = \sqrt{2.56}\][/tex]
[tex]\[a \approx 1.60\][/tex]
### 14. [tex]\(a = 1.2, c = 3.7\)[/tex]
To find [tex]\(b\)[/tex], we use:
[tex]\[b = \sqrt{c^2 - a^2}\][/tex]
Substituting the given values:
[tex]\[b = \sqrt{3.7^2 - 1.2^2}\][/tex]
[tex]\[b = \sqrt{13.69 - 1.44}\][/tex]
[tex]\[b = \sqrt{12.25}\][/tex]
[tex]\[b \approx 3.50\][/tex]
