Question

Graph the points. Decide whether they are vertices of a right triangle. $$(5,4),(2,1),(-3,2)$$

Short answer

By calculating the distances between the points, applying these distances to the Pythagorean theorem and checking for compliance, we can determine whether the given points form a right triangle or not. Whether they do indeed form such a triangle will be revealed by the calculations and cannot be stated without.

Step-by-step solution

Plot the points

Plot the points \( (5,4), (2,1), \) and \( (-3,2) \) on the Cartesian coordinate system to represent the potential vertices of the triangle.

Calculate distances

Use the distance formula, \(d=\sqrt{{(x_2-x_1)^2+(y_2-y_1)^2}}\), to calculate the distances between all three pairs of points. That is, the distances between the points (5,4) & (2,1), (2,1) & (-3,2), and (-3,2) & (5,4). These distances represent the potential sides of the triangle.

Apply the Pythagorean theorem

With the three distances calculated, label the longest distance as 'c'. The other two shorter distances will be labelled as 'a' and 'b'. Apply the Pythagorean Theorem formula \(a^2 + b^2 = c^2\) to check if these distances satisfy the theorem. The triangle will be a right triangle if the formula is satisfied, i.e., if \(a^2 + b^2\) equals to \(c^2\) or ca be approximated to it.