Question

A triangle has vertices B(2, 4), R(0, 6), and O(3, 6). Show calculations of the length of each side and tell whether the triangle is a scalene triangle, an isosceles triangle or an equilateral triangle. Note: Saying that you graphed it will not be a point-earning solution, in this case. I need to see that you can execute the mathematical processes taught in the unit. (4 points)

Answer 1

To determine the type of triangle formed by the vertices B(2, 4), R(0, 6), and O(3, 6), we first need to find the lengths of each side using the distance formula. The distance \( d \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

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

### Step 1: Calculate the lengths of each side.

**1. Length of side BR:**

Vertices B(2, 4) and R(0, 6):

\[
BR = \sqrt{(0 - 2)^2 + (6 - 4)^2}
\]
\[
= \sqrt{(-2)^2 + (2)^2}
\]
\[
= \sqrt{4 + 4}
\]
\[
= \sqrt{8}
\]
\[
= 2\sqrt{2}
\]

**2. Length of side RO:**

Vertices R(0, 6) and O(3, 6):

\[
RO = \sqrt{(3 - 0)^2 + (6 - 6)^2}
\]
\[
= \sqrt{(3)^2 + (0)^2}
\]
\[
= \sqrt{9}
\]
\[
= 3
\]

**3. Length of side OB:**

Vertices O(3, 6) and B(2, 4):

\[
OB = \sqrt{(2 - 3)^2 + (4 - 6)^2}
\]
\[
= \sqrt{(-1)^2 + (-2)^2}
\]
\[
= \sqrt{1 + 4}
\]
\[
= \sqrt{5}
\]

### Step 2: Summarize the lengths of the sides.

- Length of side BR = \( 2\sqrt{2} \)
- Length of side RO = \( 3 \)
- Length of side OB = \( \sqrt{5} \)

### Step 3: Determine the type of triangle.

To classify the triangle, we compare the lengths of the sides:

- \( 2\sqrt{2} \approx 2.83 \)
- \( 3 \)
- \( \sqrt{5} \approx 2.24 \)

Since all three lengths are different, the triangle is classified as a **scalene triangle**.

### Conclusion:

The triangle formed by the vertices B(2, 4), R(0, 6), and O(3, 6) is a **scalene triangle** since all sides have different lengths.