Three points A(1, –2), B(3, 4) and C(4, 7) forma)A straight lineb)An equilateral trianglec)A right-angled triangled)None of theseCorrect answer is option 'A'. Can you explain this answer?

Question

Three points A(1, &ndash;2), B(3, 4) and C(4, 7) forma)A straight lineb)An equilateral trianglec)A right-angled triangled)None of theseCorrect answer is option 'A'. Can you explain this answer?

Answer

Distance from A to B =

Distance from B to C =

Distance from C to A =

Since, AB + BC = AC
∴ Points lie on a straight line.

Answer

Understanding the Points A, B, and C
To determine the relationship between the points A(1, –2), B(3, 4), and C(4, 7), we need to analyze their coordinates and check if they are collinear (form a straight line).
Calculating Slopes
The slope of a line through two points (x1, y1) and (x2, y2) can be calculated using the formula:
Slope (m) = (y2 - y1) / (x2 - x1)
1. Slope between A and B:
- Points A(1, -2) and B(3, 4):
- m_AB = (4 - (-2)) / (3 - 1) = 6 / 2 = 3
2. Slope between B and C:
- Points B(3, 4) and C(4, 7):
- m_BC = (7 - 4) / (4 - 3) = 3 / 1 = 3
3. Slope between A and C:
- Points A(1, -2) and C(4, 7):
- m_AC = (7 - (-2)) / (4 - 1) = 9 / 3 = 3
Conclusion on Collinearity
- Since the slopes m_AB, m_BC, and m_AC are equal (all equal to 3), the points A, B, and C are collinear.
- This means they lie on the same straight line, confirming option 'A'.
Final Evaluation
- Option A: Straight Line - Correct
- Option B: Equilateral Triangle - Incorrect (not all sides are equal)
- Option C: Right-angled Triangle - Incorrect (no right angle formed)
- Option D: None of these - Incorrect
Thus, the conclusion that the points A, B, and C form a straight line is accurate.

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