The base of an equilateral triangle with side $2a$ lies along the y-axis such that the mid-point of the base is at the origin. Find vertices of the triangle.

Find the distance between P ($x_1$, $y_1$) and Q ($x_2$, $y_2$) when : (i) PQ is parallel to the y-axis

Without using the Pythagoras theorem, show that the points (4, 4), (3, 5) and (–1, –1) are the vertices of a right angled triangle.

Draw a quadrilateral in the Cartesian plane, whose vertices are (– 4, 5), (0, 7), (5, – 5) and (– 4, –2). Also, find its area.

The slope of a line is double of the slope of another line. If tangent of the angle between them is $\frac{1}{3}$, find the slopes of the lines.

Find the distance between P ($x_1$, $y_1$) and Q ($x_2$, $y_2$) when : (ii) PQ is parallel to the x-axis.

Find a point on the x-axis, which is equidistant from the points (7, 6) and (3, 4).

Find the slope of a line, which passes through the origin, and the mid-point of the line segment joining the points P (0, – 4) and B (8, 0).

A line passes through ($x_1$, $y_1$) and ($h$, $k$). If slope of the line is $m$, show that $k – y_1 = m (h – x_1)$.

Find the slope of the line, which makes an angle of $30^\circ$ with the positive direction of y-axis measured anticlockwise.

Without using distance formula, show that points (– 2, – 1), (4, 0), (3, 3) and (–3, 2) are the vertices of a parallelogram.

Find the angle between the x-axis and the line joining the points (3,–1) and (4,–2).