Question

determine whether a triangle with the given side lengths is a right triangle.
side lengths \t(a) 3, 9, 10 \t(b) 5, 11, 12 \t(c) 11, 13, 17 \t(d) 27, 36, 45
right triangle \t○ \t○ \t○ \t○
not a right triangle \t○ \t○ \t○ \t○
not enough information \t○ \t○ \t○ \t○

Explanation:

To determine if a triangle with given side lengths is a right triangle, we use the Pythagorean theorem, which states that for a right triangle with side lengths \(a\), \(b\), and hypotenuse \(c\) (where \(c\) is the longest side), \(a^{2}+b^{2}=c^{2}\) must hold.

Part (a): Side lengths 3, 9, 10

Step 1: Identify the longest side

The longest side is 10. So, we check if \(3^{2}+9^{2}=10^{2}\).

Step 2: Calculate each term

Calculate \(3^{2}=9\), \(9^{2}=81\), and \(10^{2}=100\).

Step 3: Check the equation

Now, \(3^{2}+9^{2}=9 + 81=90\), and \(10^{2}=100\). Since \(90
eq100\), the triangle with side lengths 3, 9, 10 is not a right triangle.

Part (b): Side lengths 5, 11, 12

Step 1: Identify the longest side

The longest side is 12. So, we check if \(5^{2}+11^{2}=12^{2}\).

Step 2: Calculate each term

Calculate \(5^{2}=25\), \(11^{2}=121\), and \(12^{2}=144\).

Step 3: Check the equation

Now, \(5^{2}+11^{2}=25 + 121 = 146\), and \(12^{2}=144\). Since \(146
eq144\), the triangle with side lengths 5, 11, 12 is not a right triangle.

Part (c): Side lengths 11, 13, 17

Step 1: Identify the longest side

The longest side is 17. So, we check if \(11^{2}+13^{2}=17^{2}\).

Step 2: Calculate each term

Calculate \(11^{2}=121\), \(13^{2}=169\), and \(17^{2}=289\).

Step 3: Check the equation

Now, \(11^{2}+13^{2}=121+169 = 290\), and \(17^{2}=289\). Since \(290
eq289\), the triangle with side lengths 11, 13, 17 is not a right triangle.

Part (d): Side lengths 27, 36, 45

Step 1: Identify the longest side

The longest side is 45. So, we check if \(27^{2}+36^{2}=45^{2}\).

Step 2: Calculate each term

Calculate \(27^{2}=729\), \(36^{2}=1296\), and \(45^{2}=2025\).

Step 3: Check the equation

Now, \(27^{2}+36^{2}=729 + 1296=2025\), and \(45^{2}=2025\). Since \(27^{2}+36^{2}=45^{2}\), the triangle with side lengths 27, 36, 45 is a right triangle.

Answer:

s:
(a) Not a right triangle
(b) Not a right triangle
(c) Not a right triangle
(d) Right triangle