Question

Let A(α, -2), B(α, 6) and C(\(\frac{\alpha}{4}\), -2) be vertices of a ΔABC. If (5, \(\frac{\alpha}{4}\)) is the circumcentre of ΔABC, then which of the following is NOT correct about ΔABC?

• A. Area is 24
• B. Perimeter is 25
• C. Circumradius is 5
• D. Inradius is 2

Answer 1

The Correct Option is B

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(\(<\alpha/4\), -2). The circumcentre is given as (5, \(\alpha/4\)). 

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, \(\alpha/4\)) as the circumcentre:

• Distance from (5, \(\alpha/4\)) to A(α, -2):
• Distance from (5, \(\alpha/4\)) to B(α, 6):
• Distance from (5, \(\alpha/4\)) to C(\(\frac{\alpha}{4}\), -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.

Answer 2

Circumcentre of ΔABC
=(\(\alpha+\frac{\frac{\alpha}{4}}{2}\),\(\frac{6-2}{2}\))
=(\(\frac{5\alpha}{8}\), 2)
=(5,\(\frac{\alpha}{4}\))
⇒α=8
Area (ΔABC)=\(\frac{1}{2}\)⋅3α4×8=24 sq. units
Perimeter =8+\(\frac{3\alpha}{4}\)+\(\sqrt{82}\)+(\(\frac{3\alpha}{4}^2\))
=8+6+10=24
Circumradius=\(\frac{10}{2}\)=5
r=\(\frac{\Delta}{s}\)
r=\(\frac{24}{12}\)
r=2
So, the correct option is (B): Perimeter is 25.