The points `A(12,8),B(-2,6)` and C(6,0) are the vertices of

Options

A. right angled `Delta`
B. isosceles `Delta`
C. equilateral `Delta`
D. None

Answer

To determine the type of triangle formed by the points A(12, 8), B(-2, 6), and C(6, 0), we will calculate the lengths of the sides using the distance formula. The distance formula between two points (x1, y1) and (x2, y2) is given by:

\[ d = \sqrt{(x2 - x1)^2 + (y2 - y1)^2} \]

### Step 1: Calculate the length of side AB
Using points A(12, 8) and B(-2, 6):

\[ AB = \sqrt{(12 - (-2))^2 + (8 - 6)^2} \]
\[ = \sqrt{(12 + 2)^2 + (8 - 6)^2} \]
\[ = \sqrt{(14)^2 + (2)^2} \]
\[ = \sqrt{196 + 4} \]
\[ = \sqrt{200} \]
\[ = 10\sqrt{2} \]

### Step 2: Calculate the length of side BC
Using points B(-2, 6) and C(6, 0):

\[ BC = \sqrt{(6 - (-2))^2 + (0 - 6)^2} \]
\[ = \sqrt{(6 + 2)^2 + (-6)^2} \]
\[ = \sqrt{(8)^2 + (6)^2} \]
\[ = \sqrt{64 + 36} \]
\[ = \sqrt{100} \]
\[ = 10 \]

### Step 3: Calculate the length of side AC
Using points A(12, 8) and C(6, 0):

\[ AC = \sqrt{(6 - 12)^2 + (0 - 8)^2} \]
\[ = \sqrt{(-6)^2 + (-8)^2} \]
\[ = \sqrt{36 + 64} \]
\[ = \sqrt{100} \]
\[ = 10 \]

### Step 4: Determine the type of triangle
Now we have the lengths of the sides:
- \( AB = 10\sqrt{2} \)
- \( BC = 10 \)
- \( AC = 10 \)

To check if triangle ABC is a right triangle, we can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.

Here, we can compare:
\[ (10\sqrt{2})^2 = 200 \]
\[ 10^2 + 10^2 = 100 + 100 = 200 \]

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

### Conclusion
The points A(12, 8), B(-2, 6), and C(6, 0) are the vertices of a right triangle.

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