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). $$ P_{1}=(-2,-1) ; \quad P_{2}=(0,7) ; \quad P_{3}=(3,2) $$

Short answer

The triangle is isosceles and right. The sides are \(2\sqrt{17}, \sqrt{34}, \sqrt{34}\).

Step-by-step solution

- Find the Distance Between Two Points

To determine the length of each side of the triangle, use the distance formula for two points \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

- Calculate Distance P1 to P2

Using the points \(P_{1} = (-2, -1)\) and \(P_{2} = (0, 7)\), the distance is: \[ d_{P_1P_2} = \sqrt{(0 - (-2))^2 + (7 - (-1))^2} = \sqrt{2^2 + 8^2} = \sqrt{4 + 64} = \sqrt{68} = 2\sqrt{17} \]

- Calculate Distance P2 to P3

Using the points \(P_{2} = (0, 7)\) and \(P_{3} = (3, 2)\), the distance is: \[ d_{P_2P_3} = \sqrt{(3 - 0)^2 + (2 - 7)^2} = \sqrt{3^2 + (-5)^2} = \sqrt{9 + 25} = \sqrt{34} \]

- Calculate Distance P1 to P3

Using the points \(P_{1} = (-2, -1)\) and \(P_{3} = (3, 2)\), the distance is: \[ d_{P_1P_3} = \sqrt{(3 - (-2))^2 + (2 - (-1))^2} = \sqrt{5^2 + 3^2} = \sqrt{25 + 9} = \sqrt{34} \]

- Compare the Side Lengths

Now compare the side lengths calculated: \[ d_{P_1P_2} = 2\sqrt{17}, \quad d_{P_2P_3} = \sqrt{34}, \quad d_{P_1P_3} = \sqrt{34} \] Since \(d_{P_2P_3} = d_{P_1P_3}\), the triangle is isosceles.

- Check for Right Triangle

For a right triangle, the Pythagorean theorem must hold, i.e., \(a^2 + b^2 = c^2\). Check using the longest side \(d_{P_1P_2}\) and the other two sides: \[ (2\sqrt{17})^2 = (\sqrt{34})^2 + (\sqrt{34})^2 \ 68 = 34 + 34 \] The equation holds true, so the triangle is a right triangle.

- Conclude the Triangle Type

The triangle is both isosceles and a right triangle, as it has two sides of equal length and satisfies the Pythagorean theorem.

Key concepts

Distance Formula

To determine the length of each side of a triangle, we use the distance formula for two points. The formula is: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
The formula helps us find the straight-line distance between any two points in a coordinate plane.
For instance, using the points \(P_{1} = (-2, -1)\) and \(P_{2} = (0, 7)\): \[ d_{P_1P_2} = \sqrt{(0 - (-2))^2 + (7 - (-1))^2} = \sqrt{2^2 + 8^2} = \sqrt{4 + 64} = \sqrt{68}= 2\sqrt{17} \]
This formula will be applied to all pairs of given points to determine the side lengths of the triangle.

Isosceles Triangle

An isosceles triangle is a triangle where at least two sides are of equal length.
To check if a triangle is isosceles, we simply compare the lengths of its sides.
Using our example, we calculated distances: \[ d_{P_1P_2} = 2\sqrt{17}, \quad d_{P_2P_3} = \sqrt{34}, \quad d_{P_1P_3} = \sqrt{34} \]
Since \(d_{P_2P_3} = d_{P_1P_3}\), the triangle has two equal sides, making it an isosceles triangle.

Right Triangle

A right triangle has one angle exactly equal to 90 degrees. To check for a right triangle, the Pythagorean theorem must hold true.
The theorem states: \[ a^2 + b^2 = c^2 \]
where \(c\) is the hypotenuse (the longest side).
For our triangle, we found: \[ d_{P_1P_2} = 2\sqrt{17}, \quad d_{P_2P_3} = \sqrt{34}, \quad d_{P_1P_3} = \sqrt{34} \]
Using \(c = 2\sqrt{17}\), we check: \[ (2\sqrt{17})^2 = (\sqrt{34})^2 + (\sqrt{34})^2 \] This evaluates to: \[ 68 = 34 + 34 \] The theorem holds true, confirming the triangle is a right triangle.

Pythagorean Theorem

The Pythagorean theorem is crucial for identifying right triangles. It states that for any right triangle: \[ a^2 + b^2 = c^2 \]
Here, \(a\) and \(b\) are the legs of the triangle, and \(c\) is the hypotenuse.
In our case, the sides are: \[ a = \sqrt{34}, \quad b = \sqrt{34}, \quad c = 2\sqrt{17} \]
To verify the Pythagorean theorem: \[ (\sqrt{34})^2 + (\sqrt{34})^2 = (2\sqrt{17})^2 \] This simplifies to: \[ 34 + 34 = 68 \]
As the values on both sides match, the side lengths confirm the triangle adheres to the Pythagorean theorem, and thus, it is a right triangle.