Question

Let \( ABC \) be an equilateral triangle. If the coordinates of \( A \) are \( (-2,2) \) and the side \( BC \) is along the line \( x + y = 6 \), then the length of the side of the triangle is

• A. \(2\sqrt{3} \)
• B. \(3\sqrt{2} \)
• C. \(4\sqrt{6} \)
• D. \(6\sqrt{6} \)
• E. \(2\sqrt{6} \)

Hint:

Distance from vertex to opposite side gives height in equilateral triangle problems.

Answer 1

Answer: (E)

Step 1: Understanding the Concept:
The problem asks for the side length of an equilateral triangle, given one vertex and the line containing the opposite side. The key is to relate the side length of an equilateral triangle to its altitude (height). The altitude is the perpendicular distance from the given vertex to the given line.
Step 2: Key Formula or Approach:
1. Calculate the altitude `h` of the triangle, which is the perpendicular distance from point A(-2, 2) to the line `x + y - 6 = 0`. The formula for the distance from a point \((x_1, y_1)\) to a line \(Ax + By + C = 0\) is \(d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}\). 2. For an equilateral triangle with side length `s` and altitude `h`, the relationship is \(h = s \frac{\sqrt{3}}{2}\). 3. Solve for `s` using the calculated value of `h`.
Step 3: Detailed Explanation:
1. Calculate the altitude (h). The point is A = (-2, 2) and the line is `x + y - 6 = 0`. Here, \(A=1, B=1, C=-6, x_1=-2, y_1=2\). \[ h = \frac{|(1)(-2) + (1)(2) - 6|}{\sqrt{1^2 + 1^2}} \] \[ h = \frac{|-2 + 2 - 6|}{\sqrt{1 + 1}} = \frac{|-6|}{\sqrt{2}} = \frac{6}{\sqrt{2}} \] To rationalize the denominator, multiply the numerator and denominator by \(\sqrt{2}\): \[ h = \frac{6\sqrt{2}}{2} = 3\sqrt{2} \] 2. Relate altitude to side length and solve for s. For an equilateral triangle: \[ h = s \frac{\sqrt{3}}{2} \] We can rearrange this to solve for `s`: \[ s = \frac{2h}{\sqrt{3}} \] Substitute the value of `h` we found: \[ s = \frac{2(3\sqrt{2})}{\sqrt{3}} = \frac{6\sqrt{2}}{\sqrt{3}} \] Rationalize the denominator by multiplying the numerator and denominator by \(\sqrt{3}\): \[ s = \frac{6\sqrt{2} \cdot \sqrt{3}}{\sqrt{3} \cdot \sqrt{3}} = \frac{6\sqrt{6}}{3} = 2\sqrt{6} \] Step 4: Final Answer:
The length of the side of the triangle is \(2\sqrt{6}\).