To find the coordinates of point E, which divides the line segment CH in a 3:1 ratio (meaning E is three-fourths of the way from C to H), we use the section formula.
The section formula for internal division is as follows:
If a line segment with endpoints A(x₁, y₁) and B(x₂, y₂) is divided by a point P in the ratio m:n, where m is the ratio of the AP segment and n is the ratio of the PB segment, then the coordinates (x, y) of P are given by:x = (mx₂ + nx₁) / (m + n)
y = (my₂ + ny₁) / (m + n)
In this case, point C has coordinates C(2, 1), point H has coordinates H(-2, -15), and the ratio of division by point E is 3:1, so we have:m = 3 (for CE)n = 1 (for EH)Now we'll apply the section formula to find the coordinates of E:E_x = (m * H_x + n * C_x) / (m + n)E_y = (m * H_y + n * C_y) / (m + n)Substitute the given values into the formula:E_x = (3 * (-2) + 1 * 2) / (3 + 1)E_y = (3 * (-15) + 1 * 1) / (3 + 1)Calculate the x-coordinate:E_x = (-6 + 2) / 4E_x = -4 / 4E_x = -1Calculate the y-coordinate:
E_y = (-45 + 1) / 4E_y = -44 / 4E_y = -11Therefore, the coordinates of point E are (-1, -11).