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Understanding how to measure the distance between two points on a coordinate plane is essential when it comes to analyzing shapes like triangles, especially when determining whether a triangle is right-angled or not. The distance formula is derived from the Pythagorean theorem and is expressed as
\( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \),
where \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of the two points. By plugging in the x and y values of the given points, you can calculate the exact distance between them.

Using the Distance Formula

To illustrate, if we're given two points A and B where A is at \( (-5,2) \) and B at \( (-2,5) \) on a coordinate plane, the distance between them can be found by plugging the values into the formula:
\(AB = \sqrt{(-2 - (-5))^2 + (5 - 2)^2} \).
This calculation gives us the length of one side of the triangle in question. Repeating this process for the other two sides will provide all three side lengths necessary to determine the shape of the triangle.

The Pythagorean theorem is a cornerstone of geometry, especially when dealing with right triangles. This theorem can be stated as:
\(a^2 + b^2 = c^2\),
where \(a\) and \(b\) are the lengths of the legs (the shorter two sides) and \(c\) is the length of the hypotenuse (the longest side opposite the right angle).

Applying the Pythagorean Theorem

If we ascertain that side AC is the longest side from our earlier calculations, we speculate that it could be the hypotenuse. Hence, we compare the squares of the lengths of sides AB and BC with the square of AC. If AC's length squared equals the sum of AB's length squared and BC's length squared, side AC is indeed the hypotenuse, conclusively proving that the triangle is right-angled at the vertex where sides AB and BC meet.

Calculating the side lengths of a triangle using the distance formula is a stepping stone to identifying its type. In the context of a right triangle, knowing the exact side lengths is critical to applying the Pythagorean theorem correctly.

Verifying Side Lengths

Once we have computed the lengths AB, BC, and AC, we have the necessary information to take further steps. For instance, AB is found to be \( \sqrt{18} \), BC as \( \sqrt{98} \), and AC as \( \sqrt{116} \). We must ensure the accuracy of these lengths, as even a small miscalculation can lead to an incorrect conclusion about the triangle's nature.
With confidence in our side lengths, we can move on to verification using the Pythagorean theorem. In the problem at hand, the theorem holds true thus, verifying our triangle as right-angled. To help students further understand the concepts, using graphs and visual representations of these calculations can be immensely helpful.