Determining the Type of Triangle ABC: An Analytical Approach

When analyzing the coordinates of points A, B, and C, we can determine the type of triangle formed by these points. In this article, we'll use coordinate geometry to find the lengths of the sides of the triangle and determine its type. This process will help us understand the relationship between the points and the geometric properties of the triangle.

Introduction to the Points and the Problem

Given the coordinates A(-1, 2), B(0, -5), and C(4, 7), our goal is to find the lengths of the sides of the triangle and determine its type. This will involve calculating the distances between each pair of points using the distance formula. The distance formula is given by:

Distance formula: d sqrt{(y_2 - y_1)^2 (x_2 - x_1)^2}

Calculating the Lengths of the Sides

1. Length of AB

The coordinates for point A are (-1, 2) and for point B are (0, -5). Plugging these into the distance formula:

AB sqrt{(-5 - 2)^2 (0 - (-1))^2}

AB sqrt{(-7)^2 (1)^2}

AB sqrt{49 1}

AB sqrt{50}

AB sqrt{25 times 2}

AB 5sqrt{2}

2. Length of BC

The coordinates for point B are (0, -5) and for point C are (4, 7). Plugging these into the distance formula:

BC sqrt{(7 - (-5))^2 (4 - 0)^2}

BC sqrt{(12)^2 (4)^2}

BC sqrt{144 16}

BC sqrt{160}

BC sqrt{16 times 10}

BC 4sqrt{10}

3. Length of AC

The coordinates for point A are (-1, 2) and for point C are (4, 7). Plugging these into the distance formula:

AC sqrt{(7 - 2)^2 (4 - (-1))^2}

AC sqrt{(5)^2 (5)^2}

AC sqrt{25 25}

AC sqrt{50}

AC sqrt{25 times 2}

AC 5sqrt{2}

Conclusion and Determination of the Type of Triangle

Now that we have calculated the lengths of the sides of the triangle, we can compare them to determine the type of triangle. We found that:

AB 5sqrt{2} units

BC 4sqrt{10} units

AC 5sqrt{2} units

Since two sides of the triangle (AB and AC) are equal (5sqrt{2}), we can conclude that the triangle is an isosceles triangle.

Conclusion

Using coordinate geometry, we have determined that the triangle ABC, with coordinates A(-1, 2), B(0, -5), and C(4, 7), is an isosceles triangle. This determination is based on the equality of two sides of the triangle.

Visual Aid

To visualize the points and the triangle, you can plot the points on a graph paper and connect them to form the triangle. This will provide a clear visual representation of the triangle and the relationships between its sides.

For further analysis or verification, you can use graph paper or a graphing software to plot the points and observe the shape of the triangle.