Horizontal And Vertical Lines

To solve this problem, we need to analyze the given information about the vertices and circumcentre of ΔABC.

The vertices of ΔABC are A(α, -2), B(α, 6), and C( , -2). The circumcentre is given as (5, ).

Step 1: Understanding the Circumcentre

The circumcentre of a triangle is equidistant from all the vertices. Hence, we have the following equations for the circumradius (R), considering the point (5, ) as the circumcentre:

• Distance from (5, ) to A(α, -2):
• Distance from (5, ) to B(α, 6):
• Distance from (5, ) to C( , -2):

Step 2: Equating Distances

Equating any two distances will give us the value of α. Solving these equations will lead us to α = 4.

Step 3: Calculating the Lengths

• Coordinates of A, B, and C for α = 4 are A(4, -2), B(4, 6), and C(1, -2).
• Using the distance formula, calculate the lengths of sides AB, BC, and CA:

Step 4: Verifying the Options

• Area: Use the determinant method to find the area of a triangle with the given vertices:
• Perimeter: Add the lengths of all sides:
• Circumradius: From the coordinates of circumcentre and vertex distances:
• Inradius: Calculate using the formula:

Conclusion:

The perimeter option stating "Perimeter is 25" is incorrect because the calculated perimeter is 16.

Given the line , we can parametrize it as:

Let be the foot of the perpendicular from to the line. The vector is:

Since is perpendicular to the line, we set up the dot product with the direction ratios :

Expanding and solving:

Substitute to find , , and :

Then,

Finally,