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Points A (4, 6), B (8, 10) and C (12, 14) are in the rectangular coor
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17 Dec 2024, 00:01
17 Dec 2024, 00:01
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Points A (4, 6), B (8, 10) and C (12, 14) are in the rectangular coordinate system formed by the x-axis and y-axis. Which of the following options are true?
Indicate all that apply
A. Points A, B and C lie on a same straight line
B. The slopes of any two pair of points among the three points A, B and C are same
C. Line AB is perpendicular to line BC
Indicate all that apply
A. Points A, B and C lie on a same straight line
B. The slopes of any two pair of points among the three points A, B and C are same
C. Line AB is perpendicular to line BC
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Re: Points A (4, 6), B (8, 10) and C (12, 14) are in the rectangular coor
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17 Dec 2024, 08:13
17 Dec 2024, 08:13
Carcass GreenlightTestPrep I am confused on this, how can A and C both be correct?
Points A (4, 6), B (8, 10) and C (12, 14) are in the rectangular coor
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17 Dec 2024, 10:17
17 Dec 2024, 10:17
Expert Reply
We know that the formula to calculate the area of triangle with vertices $\(\left(\mathrm{x}_1, \mathrm{y}_1\right),\left(\mathrm{x}_2, \mathrm{y}_2\right) \&\left(\mathrm{x}_3, \mathrm{y}_3\right)\)$ is
$\(\frac{1}{2}\left|\mathrm{x}_1\left(\mathrm{y}_2-\mathrm{y}_3\right)+\mathrm{x}_2\left(\mathrm{y}_3-\mathrm{y}_1\right)+\mathrm{x}_3\left(\mathrm{y}_1-\mathrm{y}_2\right)\right|\)$
Using the above formula, the area of the triangle, if any, formed by the points $\(A(4,6), B(8,10)\)$ $\(\& C(12,14)$ is $\frac{1}{2}|4(10-14)+8(14-6)+12(6-10)|=\frac{1}{2}|-16+64-48|=\frac{1}{2} \times 0=0\)$
The area being zero means the three points $\(\mathrm{A}, \mathrm{B} \& \mathrm{C}\)$ are collinear i.e. in a straight line.
Since the points are on a straight line, the slope found using any two of the three points should be the same.
Hence all three options (A), (B) \& (C) are correct.
$\(\frac{1}{2}\left|\mathrm{x}_1\left(\mathrm{y}_2-\mathrm{y}_3\right)+\mathrm{x}_2\left(\mathrm{y}_3-\mathrm{y}_1\right)+\mathrm{x}_3\left(\mathrm{y}_1-\mathrm{y}_2\right)\right|\)$
Using the above formula, the area of the triangle, if any, formed by the points $\(A(4,6), B(8,10)\)$ $\(\& C(12,14)$ is $\frac{1}{2}|4(10-14)+8(14-6)+12(6-10)|=\frac{1}{2}|-16+64-48|=\frac{1}{2} \times 0=0\)$
The area being zero means the three points $\(\mathrm{A}, \mathrm{B} \& \mathrm{C}\)$ are collinear i.e. in a straight line.
Since the points are on a straight line, the slope found using any two of the three points should be the same.
Hence all three options (A), (B) \& (C) are correct.
