Question

If \( -1 + 7i \), \( -1 + xi \) and \( 3 + 3i \) are the three vertices of an isosceles triangle which is right angled at \( -1 + xi \), then the value of \( x \) is equal to

• A. \( -1 \)
• B. \( 3 \)
• C. \( -3 \)
• D. \( 7 \)
• E. \( -7 \)

Hint:

A quick sketch of the points in the complex plane can often reveal geometric properties like vertical or horizontal lines, saving you from lengthy distance or slope formula calculations.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
Vertices of a triangle given as complex numbers can be mapped directly to coordinates \( (a, b) \) on the Cartesian (Argand) plane.
The problem involves finding an unknown coordinate based on the properties of a right-angled isosceles triangle.

Step 2: Key Formula or Approach:
Convert the complex numbers to coordinate points: \( A(-1, 7) \), \( B(-1, x) \), and \( C(3, 3) \).
Use geometric properties: The angle at \( B \) is \( 90^\circ \), meaning line segments \( AB \) and \( BC \) are perpendicular.
Also, since it is isosceles, the lengths of the legs adjacent to the right angle must be equal (i.e., \( AB = BC \)).

Step 3: Detailed Explanation:
Let the vertices in the coordinate plane be:
\( A = (-1, 7) \)
\( B = (-1, x) \)
\( C = (3, 3) \)
The triangle is right-angled at \( B(-1, x) \).
Notice that the x-coordinates of points \( A \) and \( B \) are both \( -1 \).
This implies that the line passing through \( A \) and \( B \) is a vertical line parallel to the y-axis.
For angle \( B \) to be \( 90^\circ \), the line passing through \( B \) and \( C \) must be perpendicular to \( AB \).
Since \( AB \) is a vertical line, \( BC \) must be a completely horizontal line parallel to the x-axis.
A horizontal line has a constant y-coordinate for all its points.
Therefore, the y-coordinate of \( B \) must equal the y-coordinate of \( C \).
This gives us: \( x = 3 \).
Let us verify if the triangle is isosceles with \( x = 3 \):
The length of segment \( AB \) is the distance between \( (-1, 7) \) and \( (-1, 3) \):
\( AB = |7 - 3| = 4 \) units.
The length of segment \( BC \) is the distance between \( (-1, 3) \) and \( (3, 3) \):
\( BC = |3 - (-1)| = 4 \) units.
Since \( AB = BC = 4 \), the triangle is indeed an isosceles right-angled triangle.

Step 4: Final Answer:
The value of \( x \) is \( 3 \).