If one vertex of equilateral `Delta` is at A (3, 4) and the base BC is x + y - 5 = 0, then the length of each side of the `Delta` is

Options

A. `3sqrt(3)`
B. `(4sqrt(3))/(5)`
C. `(2sqrt(2))/(3)`
D. `2sqrt(2)`

Answer

To find the length of each side of the equilateral triangle ABC, given that one vertex A is at (3, 4) and the base BC lies on the line x + y - 5 = 0, we can follow these steps:

### Step 1: Identify the line equation
The equation of the line BC is given as:
\[ x + y - 5 = 0 \]
This can be rewritten in the standard form:
\[ x + y = 5 \]

### Step 2: Calculate the perpendicular distance from point A to line BC
To find the length of the height AD from point A to line BC, we use the formula for the perpendicular distance from a point (x1, y1) to a line Ax + By + C = 0:
\[ \text{Distance} = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \]

Here, A = 1, B = 1, C = -5, and the coordinates of point A are (x1, y1) = (3, 4).

Substituting these values into the formula:
\[ \text{Distance} = \frac{|1(3) + 1(4) - 5|}{\sqrt{1^2 + 1^2}} \]
\[ = \frac{|3 + 4 - 5|}{\sqrt{1 + 1}} \]
\[ = \frac{|2|}{\sqrt{2}} \]
\[ = \frac{2}{\sqrt{2}} = \sqrt{2} \]

### Step 3: Relate the height to the side of the equilateral triangle
The height (h) of an equilateral triangle with side length A is given by:
\[ h = \frac{\sqrt{3}}{2} A \]

We have found that the height AD = \(\sqrt{2}\). Therefore, we can set up the equation:
\[ \sqrt{2} = \frac{\sqrt{3}}{2} A \]

### Step 4: Solve for the side length A
To find A, we rearrange the equation:
\[ A = \frac{2\sqrt{2}}{\sqrt{3}} \]

### Step 5: Simplify the expression for A
To simplify \( A \):
\[ A = \frac{2\sqrt{2}}{\sqrt{3}} = \frac{2\sqrt{6}}{3} \]

### Step 6: Final answer
Thus, the length of each side of the equilateral triangle ABC is:
\[ A = \frac{2\sqrt{6}}{3} \]