Proving Points Form a Right Triangle: Using Distance and Slopes

In geometrical problem solving, determining whether a set of points form a right triangle is a common task. This is particularly useful in various fields such as engineering, physics, and computer graphics.

Introduction to the Problem

The given points are: ((-1, 3)), ((0, 5)), and ((3, 1)). We need to show that these points form a right triangle. To do this, we will use the distance formula and the Pythagorean theorem, and also explore the slopes of the lines connecting the points.

Step 1: Calculate the Distances

We start by calculating the distances between each pair of points using the distance formula:

Calculate the distance between ((-1, 3)) and ((0, 5)):

d_1  sqrt{(0 - (-1))^2   (5 - 3)^2}  sqrt{1^2   2^2}  sqrt{5}

Calculate the distance between ((0, 5)) and ((3, 1)):

d_2  sqrt{(3 - 0)^2   (1 - 5)^2}  sqrt{3^2   (-4)^2}  sqrt{9   16}  sqrt{25}  5

Calculate the distance between ((-1, 3)) and ((3, 1)):

d_3  sqrt{(3 - (-1))^2   (1 - 3)^2}  sqrt{4^2   (-2)^2}  sqrt{16   4}  sqrt{20}  2sqrt{5}

The lengths of the sides of the triangle are:

d_1 sqrt{5} d_2 5 d_3 2sqrt{5}

Step 2: Identify the Longest Side

The longest side is clearly d_2 5.

Step 3: Verify the Pythagorean Theorem

To verify if the triangle is a right triangle, we check if the square of the longest side (hypotenuse) is equal to the sum of the squares of the other two sides. This is the Pythagorean theorem.

Calculate the squares of the sides:

d_1^2 (sqrt{5})^2 5 d_2^2 5^2 25 d_3^2 (2sqrt{5})^2 4 cdot 5 20

Substitute these values into the Pythagorean theorem:

d_2^2  d_1^2   d_3^2 implies 25  5   20
25  25

Since the equation holds true, the points ((-1, 3)), ((0, 5)), and ((3, 1)) form a right triangle.

Step 4: Using Slopes to Confirm Perpendicularity

Alternatively, we can confirm the right angle using the slopes of the lines connecting the points.

Slope of line AB (from ((-1, 3)) to ((0, 5))):

text{slope of AB}  frac{5 - 3}{0 - (-1)}  frac{2}{1}  2

Slope of line BC (from ((0, 5)) to ((3, 1))):

text{slope of BC}  frac{1 - 5}{3 - 0}  frac{-4}{3}

Slope of line AC (from ((-1, 3)) to ((3, 1))):

text{slope of AC}  frac{1 - 3}{3 - (-1)}  frac{-2}{4}  -frac{1}{2}

The slope of AB is 2, and the slope of AC is (-frac{1}{2}). These two slopes are negative reciprocals of each other, which means the lines are perpendicular. Thus, the triangle formed by the points is a right triangle, and the right angle is at point B.

Conclusion

By using both the distance formula and the slopes of the lines, we have shown that the points ((-1, 3)), ((0, 5)), and ((3, 1)) are the vertices of a right triangle. This method is both intuitive and practical, making it a valuable tool in geometric problem-solving.