Question

in exercises 9 - 14, tell whether the triangle is a right triangle. example 4
9.
10.
11.
12.
13.
14.

Explanation:

Step1: Recall the Pythagorean theorem

For a triangle with side lengths \(a\), \(b\), and \(c\) (\(c\) is the longest side), if \(a^{2}+b^{2}=c^{2}\), then the triangle is a right - triangle.

Step2: Check for triangle in Exercise 9

Let \(a = 65\), \(b=72\), \(c = 97\).
Calculate \(a^{2}+b^{2}\):
\(a^{2}+b^{2}=65^{2}+72^{2}=4225 + 5184=9409\)
Calculate \(c^{2}\):
\(c^{2}=97^{2}=9409\)
Since \(a^{2}+b^{2}=c^{2}\), the triangle in Exercise 9 is a right - triangle.

Step3: Check for triangle in Exercise 10

Let \(a = 11.4\), \(b = 21.2\), \(c = 23\)
Calculate \(a^{2}+b^{2}\):
\(a^{2}+b^{2}=11.4^{2}+21.2^{2}=129.96+449.44 = 579.4\)
Calculate \(c^{2}\):
\(c^{2}=23^{2}=529\)
Since \(a^{2}+b^{2}
eq c^{2}\), the triangle in Exercise 10 is not a right - triangle.

Step4: Check for triangle in Exercise 11

Let \(a = 10\), \(b = 14\), \(c = 4\sqrt{19}\)
Calculate \(a^{2}+b^{2}\):
\(a^{2}+b^{2}=10^{2}+14^{2}=100 + 196=296\)
Calculate \(c^{2}\):
\(c^{2}=(4\sqrt{19})^{2}=16\times19 = 304\)
Since \(a^{2}+b^{2}
eq c^{2}\), the triangle in Exercise 11 is not a right - triangle.

Step5: Check for triangle in Exercise 12

Let \(a = 1\), \(b = 5\), \(c=\sqrt{26}\)
Calculate \(a^{2}+b^{2}\):
\(a^{2}+b^{2}=1^{2}+5^{2}=1 + 25=26\)
Calculate \(c^{2}\):
\(c^{2}=(\sqrt{26})^{2}=26\)
Since \(a^{2}+b^{2}=c^{2}\), the triangle in Exercise 12 is a right - triangle.

Step6: Check for triangle in Exercise 13

Let \(a = 2\), \(b = 6\), \(c = 3\sqrt{5}\)
Calculate \(a^{2}+b^{2}\):
\(a^{2}+b^{2}=2^{2}+6^{2}=4 + 36=40\)
Calculate \(c^{2}\):
\(c^{2}=(3\sqrt{5})^{2}=9\times5 = 45\)
Since \(a^{2}+b^{2}
eq c^{2}\), the triangle in Exercise 13 is not a right - triangle.

Step7: Check for triangle in Exercise 14

Let \(a = 39\), \(b = 80\), \(c = 89\)
Calculate \(a^{2}+b^{2}\):
\(a^{2}+b^{2}=39^{2}+80^{2}=1521+6400 = 7921\)
Calculate \(c^{2}\):
\(c^{2}=89^{2}=7921\)
Since \(a^{2}+b^{2}=c^{2}\), the triangle in Exercise 14 is a right - triangle.

Answer:

• Exercise 9: Right - triangle
• Exercise 10: Not a right - triangle
• Exercise 11: Not a right - triangle
• Exercise 12: Right - triangle
• Exercise 13: Not a right - triangle
• Exercise 14: Right - triangle