The triangle joinilng the points A(2,7),B(4,-1),C(-2,6) is

Options

A. equilateral
B. Right angled
C. isosceles
D. None of these

Answer

To determine the type of triangle formed by the points A(2,7), B(4,-1), and C(-2,6), we will calculate the lengths of the sides using the distance formula and then check if it satisfies the Pythagorean theorem to identify if it is a right triangle.

### Step 1: Calculate the length of side AB
Using the distance formula:
\[ AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
For points A(2,7) and B(4,-1):
\[ AB = \sqrt{(4 - 2)^2 + (-1 - 7)^2} = \sqrt{(2)^2 + (-8)^2} = \sqrt{4 + 64} = \sqrt{68} \]

### Step 2: Calculate the length of side BC
For points B(4,-1) and C(-2,6):
\[ BC = \sqrt{(-2 - 4)^2 + (6 - (-1))^2} = \sqrt{(-6)^2 + (7)^2} = \sqrt{36 + 49} = \sqrt{85} \]

### Step 3: Calculate the length of side AC
For points A(2,7) and C(-2,6):
\[ AC = \sqrt{(-2 - 2)^2 + (6 - 7)^2} = \sqrt{(-4)^2 + (-1)^2} = \sqrt{16 + 1} = \sqrt{17} \]

### Step 4: Check if the triangle is a right triangle
To determine if the triangle is a right triangle, we will check if the square of the longest side is equal to the sum of the squares of the other two sides.

We have:
- \(AB^2 = 68\)
- \(BC^2 = 85\)
- \(AC^2 = 17\)

Now, we check:
\[ AB^2 + AC^2 = 68 + 17 = 85 = BC^2 \]

Since \(AB^2 + AC^2 = BC^2\), we conclude that triangle ABC is a right triangle.

### Final Conclusion
The triangle joining the points A(2,7), B(4,-1), and C(-2,6) is a right triangle.

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