Question

Find the length of each side of the triangle determined by the three points \(P_{1}, P_{2},\) and \(P_{3}\). State whether the triangle is an isosceles triangle, a right triangle, neither of these, or both. (An isosceles triangle is one in which at least two of the sides are of equal length). $$ P_{1}=(-1,4) ; \quad P_{2}=(6,2) ; \quad P_{3}=(4,-5) $$

Short answer

The triangle is an isosceles and right triangle because two sides are equal and the Pythagorean theorem holds.

Step-by-step solution

- Find the distance between P1 and P2

Use the distance formula: \ \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\) \ Substitute the coordinates of \(P_1\) and \(P_2\): \ \(d_{12} = \sqrt{(6 - (-1))^2 + (2 - 4)^2} = \sqrt{(7)^2 + (-2)^2} = \sqrt{49 + 4} = \sqrt{53}\)

- Find the distance between P2 and P3

Again, use the distance formula: \ \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\) \ Substitute the coordinates of \(P_2\) and \(P_3\): \ \(d_{23} = \sqrt{(4 - 6)^2 + (-5 - 2)^2} = \sqrt{(-2)^2 + (-7)^2} = \sqrt{4 + 49} = \sqrt{53}\)

- Find the distance between P1 and P3

Using the distance formula once again: \ \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\) \ Substitute the coordinates of \(P_1\) and \(P_3\): \ \(d_{13} = \sqrt{(4 - (-1))^2 + (-5 - 4)^2} = \sqrt{(5)^2 + (-9)^2} = \sqrt{25 + 81} = \sqrt{106}\)

- Identify the type of triangle

Now that we have the lengths of all the sides: \ \(d_{12} = \sqrt{53}\), \(d_{23} = \sqrt{53}\), and \(d_{13} = \sqrt{106}\), compare the lengths: \ Since \(d_{12} = d_{23}\) and they are not all equal, it is an isosceles triangle.\Check if it's a right triangle by verifying the Pythagorean theorem: \ \(\sqrt{53}^2 + \sqrt{53}^2 = 53 + 53 = 106 =\sqrt{106}^2\) Hence, the triangle is also a right triangle.

Key concepts

Distance Formula

The distance formula is crucial for finding the length of a side in a triangle when you have the coordinates of the points. This formula comes from the Pythagorean theorem. Here's the formula:
\(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)
Let's break it down.

• The numbers \(x_1, y_1\) are the coordinates of the first point.
• The numbers \(x_2, y_2\) are the coordinates of the second point.
• You subtract the \(x\) and \(y\) values separately, square them, add the results, and take the square root of the sum.

Our exercise points were \(P_1 = (-1, 4)\), \(P_2 = (6, 2)\), and \(P_3 = (4, -5)\). By applying the distance formula to find \(d_{12}, d_{23}, \text{and} d_{13}\), we calculated the distances between the points to determine the type of triangle. Repeating similar steps enables you to solve many geometric problems.
The formula helps in various practical applications like navigation, computer graphics, and more.

Isosceles Triangle

An isosceles triangle has at least two sides of equal length.
Here is how to check if a triangle is isosceles:

• Calculate the length of all three sides.
• Compare these lengths. If at least two sides are equal, then it’s an isosceles triangle.

In our exercise:

• The distances were \(d_{12} = \sqrt{53}\) and \(d_{23} = \sqrt{53}\),
• and \(d_{13} = \sqrt{106}\).

Since \(d_{12}\) and \(d_{23}\) are equal, the triangle is an isosceles triangle.Identifying an isosceles triangle is important in geometry due to its unique properties, such as symmetry, which can simplify various geometric proofs and constructions.

Right Triangle

A right triangle has one angle that is exactly 90 degrees. To determine if a triangle is a right triangle, use the Pythagorean theorem:
\(a^2 + b^2 = c^2\)
Here’s a simple step-by-step check:

• Identify the lengths of the three sides.
• Square the lengths.
• Check if the sum of the squares of two sides equals the square of the longest side.

In our exercise:

• Our side lengths are \(d_{12} = d_{23} = \sqrt{53}\) and \(d_{13} = \sqrt{106}\).
• Verify: \(\sqrt{53}^2 + \sqrt{53}^2 = 53 + 53 = 106 = \sqrt{106}^2.\)

This confirms that the triangle is a right triangle. Knowing the right triangle's properties is helpful in many fields like architecture, astronomy, and various engineering applications, because they provide a base for many calculations and designs.