Name the type of triangle formed by the points \(A(-5,6),\ B(-4,-2),\ C(7,5)\).
Scalene.
Step 1: Recall the distance formula.
The distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Step 2: Find length of side AB.
Coordinates: \(A(-5, 6), B(-4, -2)\)
\[ AB^2 = (-4 - (-5))^2 + (-2 - 6)^2 \]
\[ AB^2 = (1)^2 + (-8)^2 = 1 + 64 = 65 \]
So, \(AB = \sqrt{65}\).
Step 3: Find length of side BC.
Coordinates: \(B(-4, -2), C(7, 5)\)
\[ BC^2 = (7 - (-4))^2 + (5 - (-2))^2 \]
\[ BC^2 = (11)^2 + (7)^2 = 121 + 49 = 170 \]
So, \(BC = \sqrt{170}\).
Step 4: Find length of side CA.
Coordinates: \(C(7, 5), A(-5, 6)\)
\[ CA^2 = (-5 - 7)^2 + (6 - 5)^2 \]
\[ CA^2 = (-12)^2 + (1)^2 = 144 + 1 = 145 \]
So, \(CA = \sqrt{145}\).
Step 5: Compare the side lengths.
We have:
Since \(65 \neq 170 \neq 145\), all three sides are of different lengths.
Step 6: Check for right triangle.
If it were a right triangle, one side squared would equal the sum of the other two (Pythagoras theorem). But:
So, it is not a right triangle.
Final Conclusion: The triangle has all unequal sides and is not right-angled. Therefore, it is a scalene triangle.