Questions: Determine whether a triangle with the given vertices is a right triangle. Right triangle Not a right triangle Cannot be determined ------------ (a) A(-3,6), B(-7,12), C(-12,9) 0 0 0 (b) P(4,11), Q(2,-12), R(-8,-4) 0 0 0 (c) D(-3,1), E(1,7), F(8,0) 0 0 0

Determine whether a triangle with the given vertices is a right triangle.

Right triangle Not a right triangle Cannot be determined
------------
(a) A(-3,6), B(-7,12), C(-12,9) 0 0 0
(b) P(4,11), Q(2,-12), R(-8,-4) 0 0 0
(c) D(-3,1), E(1,7), F(8,0) 0 0 0

Transcript text: Determine whether a triangle with the given vertices is a right triangle. \begin{tabular}{|l|c|c|c|} \hline & \begin{tabular}{c} Right \\ triangle \end{tabular} & \begin{tabular}{c} Not a right \\ triangle \end{tabular} & \begin{tabular}{c} Cannot be \\ determined \end{tabular} \\ \hline (a) $A(-3,6), B(-7,12), C(-12,9)$ & 0 & 0 & 0 \\ \hline (b) $P(4,11), Q(2,-12), R(-8,-4)$ & 0 & 0 & 0 \\ \hline (c) $D(-3,1), E(1,7), F(8,0)$ & 0 & 0 & 0 \\ \hline \end{tabular}

Solution

Solution Steps

Step 1: Calculate Distances

For the vertices $ A(-3, 6) $, $ B(-7, 12) $, and $ C(-12, 9) $:

Calculate $ AB = \sqrt{((-7) - (-3))^2 + ((12) - (6))^2} = \sqrt{(-4)^2 + (6)^2} = \sqrt{16 + 36} = \sqrt{52} $.
Calculate $ BC = \sqrt{((-12) - (-7))^2 + ((9) - (12))^2} = \sqrt{(-5)^2 + (-3)^2} = \sqrt{25 + 9} = \sqrt{34} $.
Calculate $ CA = \sqrt{((-3) - (-12))^2 + ((6) - (9))^2} = \sqrt{(9)^2 + (-3)^2} = \sqrt{81 + 9} = \sqrt{90} $.

Step 2: Check Right Triangle Condition

For the sides $ AB = \sqrt{52} $, $ BC = \sqrt{34} $, and $ CA = \sqrt{90} $:

Identify the longest side, which is $ CA $.
Check if $ CA^2 = AB^2 + BC^2 $: \[ 90 = 52 + 34 \quad \text{(False)} \] Thus, triangle $ ABC $ is not a right triangle.

Step 3: Calculate Distances for Second Triangle

For the vertices $ P(4, 11) $, $ Q(2, -12) $, and $ R(-8, -4) $:

Calculate $ PQ = \sqrt{(2 - 4)^2 + (-12 - 11)^2} = \sqrt{(-2)^2 + (-23)^2} = \sqrt{4 + 529} = \sqrt{533} $.
Calculate $ QR = \sqrt{(-8 - 2)^2 + (-4 - (-12))^2} = \sqrt{(-10)^2 + (8)^2} = \sqrt{100 + 64} = \sqrt{164} $.
Calculate $ RP = \sqrt{(4 - (-8))^2 + (11 - (-4))^2} = \sqrt{(12)^2 + (15)^2} = \sqrt{144 + 225} = \sqrt{369} $.

Step 4: Check Right Triangle Condition for Second Triangle

For the sides $ PQ = \sqrt{533} $, $ QR = \sqrt{164} $, and $ RP = \sqrt{369} $:

Identify the longest side, which is $ RP $.
Check if $ RP^2 = PQ^2 + QR^2 $: \[ 369 = 533 + 164 \quad \text{(True)} \] Thus, triangle $ PQR $ is a right triangle.

Step 5: Calculate Distances for Third Triangle

For the vertices $ D(-3, 1) $, $ E(1, 7) $, and $ F(8, 0) $:

Calculate $ DE = \sqrt{(1 - (-3))^2 + (7 - 1)^2} = \sqrt{(4)^2 + (6)^2} = \sqrt{16 + 36} = \sqrt{52} $.
Calculate $ EF = \sqrt{(8 - 1)^2 + (0 - 7)^2} = \sqrt{(7)^2 + (-7)^2} = \sqrt{49 + 49} = \sqrt{98} $.
Calculate $ FD = \sqrt{(-3 - 8)^2 + (1 - 0)^2} = \sqrt{(-11)^2 + (1)^2} = \sqrt{121 + 1} = \sqrt{122} $.

Step 6: Check Right Triangle Condition for Third Triangle

For the sides $ DE = \sqrt{52} $, $ EF = \sqrt{98} $, and $ FD = \sqrt{122} $:

Identify the longest side, which is $ FD $.
Check if $ FD^2 = DE^2 + EF^2 $: \[ 122 = 52 + 98 \quad \text{(False)} \] Thus, triangle $ DEF $ is not a right triangle.