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The co-ordinates of the vertices of a triangle are [m(m + 1), (m + 1)], [(m + 1)(m + 2), (m + 2)] and [(m + 2)(m + 3), (m + 3)]. Then which one among the following is correct?
  • a)
    The area of the triangle is dependent on m
  • b)
    The area of the triangle is independent on m
  • c)
    Answer cannot be predicted
  • d)
    Data inadequate
Correct answer is option 'B'. Can you explain this answer?
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Most Upvoted Answer
The co-ordinates of the vertices of a triangle are [m(m + 1), (m + 1)]...
The area o the triangle with the given point as vertices is,


Now, by performing the row operation R2 = R2 – R1 and R3 = R3 – R2

Now, breaking the determinant we get,
= 1/2 (2m + 2 – 2m – 4)
= -1
Thus, it is independent of m.
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The co-ordinates of the vertices of a triangle are [m(m + 1), (m + 1)]...
Given:
The coordinates of the vertices of a triangle are [(m(m+1), (m+1)], [(m+1)(m+2), (m+2)] and [(m+2)(m+3), (m+3)].

To find:
Which one among the following is correct?

Solution:
To determine whether the area of the triangle is dependent or independent on 'm', we can use the concept of the determinant.

Step 1: Find the coordinates of the vertices of the triangle:
Let's find the coordinates of the vertices of the triangle using the given expressions:

Vertex A: [(m(m+1), (m+1)] = (m^2 + m, m + 1)
Vertex B: [(m+1)(m+2), (m+2)] = (m^2 + 3m + 2, m + 2)
Vertex C: [(m+2)(m+3), (m+3)] = (m^2 + 5m + 6, m + 3)

Step 2: Calculate the area of the triangle:
The area of a triangle with vertices (x1, y1), (x2, y2), and (x3, y3) can be calculated using the determinant formula as follows:

Area = 1/2 * |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|

Substituting the coordinates of the vertices, we get:

Area = 1/2 * |(m^2 + m)(m + 2 - (m + 3)) + (m^2 + 3m + 2)(m + 3 - (m + 1)) + (m^2 + 5m + 6)(m + 1 - (m + 2))|

Simplifying the expression:

Area = 1/2 * |(-m - 2) + 2(2m + 2) - (m^2 + 2m) + 2(4m + 4) - (m^2 + 2m - 2) + 2(6m + 6) - (m^2 + 2m + 2)|

Area = 1/2 * |6m + 6 - m^2 - 2m + 8m + 8 - m^2 - 2m + 12m + 12 - m^2 - 2m - 2|

Simplifying further:

Area = 1/2 * |20m + 24 - 3m^2 - 8m - 2|

Area = 1/2 * |-3m^2 + 12m + 22|

Conclusion:
From the above expression, it is clear that the area of the triangle is dependent on 'm' because the expression contains the term 'm'. Therefore, the correct answer is (a) The area of the triangle is dependent on 'm'.
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The co-ordinates of the vertices of a triangle are [m(m + 1), (m + 1)], [(m + 1)(m + 2), (m + 2)] and [(m + 2)(m + 3), (m + 3)]. Then which one among the following is correct?a)The area of the triangle is dependent on mb)The area of the triangle is independent on mc)Answer cannot be predictedd)Data inadequateCorrect answer is option 'B'. Can you explain this answer?
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