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.
Get the complete, step-by-step math solution for: "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...". Powered by SolveForX AI math tutor.
Step-by-Step Solution
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 a3/2=23/2=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−y1=x2−x1y2−y1(x−x1). For line AB, we use points B(1,0) and A(2, 3). For line AC, we use points C(3,0) and A(2, 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 (xN,yN), then M will have coordinates (xN/2,yN/2).
Step 4: Substitute M and N into line equations
Since M lies on AB, its coordinates (xM,yM) satisfy the equation of line AB. Substituting xM=xN/2 and yM=yN/2 into the equation for AB gives us a relationship between xN and yN. Since N lies on AC, its coordinates (xN,yN) must also satisfy the equation of line AC.
Step 5: Solve for N and then M
Now we have two equations for yN in terms of xN. We can equate them to solve for xN. Once xN is found, we substitute it back into either equation to find yN. Finally, we use the relationship M=(xN/2,yN/2) to find the coordinates of M.