find the length of each side of the triangledetermined by the three points P1, P2, and P3. State whether thetriangle is an isosceles triangle, a right triangle, neither of these, orboth. (An isosceles triangle is one in which at least two of the sidesare of equal length.) P1 = (2, 1); P2 = ( - 4, 1); P3 = (- 4, - 3)

To find the lengths of each side of the triangle determined by the three points \( P_1 = (2, 1) \), \( P_2 = (-4, 1) \), and \( P_3 = (-4, -3) \), we will use the distance formula:

\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

### Step 1: Calculate the lengths of the sides

**Length of side \( P_1P_2 \):**
\[
P_1P_2 = \sqrt{(-4 - 2)^2 + (1 - 1)^2} = \sqrt{(-6)^2 + 0^2} = \sqrt{36} = 6
\]

**Length of side \( P_2P_3 \):**
\[
P_2P_3 = \sqrt{(-4 - (-4))^2 + (-3 - 1)^2} = \sqrt{0^2 + (-4)^2} = \sqrt{16} = 4
\]

**Length of side \( P_3P_1 \):**
\[
P_3P_1 = \sqrt{(2 - (-4))^2 + (1 - (-3))^2} = \sqrt{(6)^2 + (4)^2} = \sqrt{36 + 16} = \sqrt{52} \approx 7.21
\]

### Step 2: Determine the type of triangle

- **Check if it is an isosceles triangle**: An isosceles triangle has at least two equal sides. In this case, \( P_1P_2 = 6 \), \( P_2P_3 = 4 \), and \( P_3P_1 \approx 7.21 \). Since none of the sides are equal, the triangle is **not isosceles**.

- **Check if it is a right triangle**: A right triangle satisfies the Pythagorean theorem, where the square of the longest side (hypotenuse) is equal to the sum of the squares of the other two sides.

Let's check:
\[
P_1P_2^2 + P_2P_3^2 = 6^2 + 4^2 = 36 + 16 = 52
\]
\[
P_3P_1^2 \approx 52
\]
Since \( P_1P_2^2 + P_2P_3^2 \approx P_3P_1^2 \), the triangle **is a right triangle**.

### Conclusion:
The triangle is a **right triangle** but **not isosceles**.

Would you like more details or have any questions?

### Related Questions:
1. How do you find the area of this right triangle?
2. What would the triangle look like on a coordinate plane?
3. Can you verify the triangle's properties using vector analysis?
4. What is the perimeter of the triangle?
5. How do the slopes of the sides confirm the triangle's type?

**Tip:** The Pythagorean theorem is a reliable method to verify if a triangle is right-angled.

Learn how to find the lengths of each side of a triangle and determine its type using coordinate geometry. This problem involves using the distance formula and the Pythagorean theorem to analyze a triangle defined by points P1(2, 1), P2(-4, 1), and P3(-4, -3). Suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation