In the coordinate plane, triangle ABC is equilateral with B(1, 0) and C(3, 0). A line through the origin O meets AB and AC at M and N respectively. If OM = MN, find the coordinates of M.

Step 1: Find coordinates of A and side length

First, we find the length of the side BC using the distance formula. Since triangle ABC is equilateral, all sides have length 2. The $x$ -coordinate of A is the midpoint of BC, which is $(1+3)/2 = 2$. The $y$ -coordinate of A is the height of an equilateral triangle with side length $a=2$, which is $a\sqrt{3}/2 = 2\sqrt{3}/2 = \sqrt{3}$.

Step 2: Determine equations of lines AB and AC

Next, we find the equations of lines AB and AC using the two-point form $y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1)$. For line AB, we use points B(1,0) and A(2, $\sqrt{3}$). For line AC, we use points C(3,0) and A(2, $\sqrt{3}$).

Step 3: Use section formula for M and N

The problem states that OM = MN. This implies that M is the midpoint of the segment ON, where O is the origin (0,0). Therefore, if N has coordinates $(x_N, y_N)$, then M will have coordinates $(x_N/2, y_N/2)$.

Step 4: Substitute M and N into line equations

Since M lies on AB, its coordinates $(x_M, y_M)$ satisfy the equation of line AB. Substituting $x_M = x_N/2$ and $y_M = y_N/2$ into the equation for AB gives us a relationship between $x_N$ and $y_N$. Since N lies on AC, its coordinates $(x_N, y_N)$ must also satisfy the equation of line AC.

Step 5: Solve for N and then M

Now we have two equations for $y_N$ in terms of $x_N$. We can equate them to solve for $x_N$. Once $x_N$ is found, we substitute it back into either equation to find $y_N$. Finally, we use the relationship $M = (x_N/2, y_N/2)$ to find the coordinates of M.

More from this chapter

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• Find the coordinates of the point which divides the line segment joining (4, -3) and (8, 5) in the ratio 3:1 internally.
• Find the relation between x and y if the point (x, y) is equidistant from (7, 1) and (3, 5).
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• Find the coordinates of the points of trisection of the line segment joining the points A(2, -2) and B(-7, 4).
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