Key Points

The distance between two points $P(x_1, y_1)$ and $Q(x_2, y_2)$ is calculated using the formula $PQ = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$. This formula is derived from the Pythagorean theorem.

A special case of the distance formula is finding the distance of a point $P(x, y)$ from the origin $O(0, 0)$. The formula simplifies to $OP = \sqrt{x^2 + y^2}$.

The coordinates of a point $P(x, y)$ that divides the line segment joining $A(x_1, y_1)$ and $B(x_2, y_2)$ internally in the ratio $m_1:m_2$ are given by $P(x, y) = \left(\frac{m_1 x_2 + m_2 x_1}{m_1 + m_2}, \frac{m_1 y_2 + m_2 y_1}{m_1 + m_2}\right)$.

The mid-point of a line segment divides it in the ratio $1:1$. The coordinates of the mid-point of the segment joining $(x_1, y_1)$ and $(x_2, y_2)$ are $\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$.

Three points A, B, and C are collinear if they lie on the same straight line. This can be verified using the distance formula: the sum of the lengths of any two segments equals the length of the third, for example, $AB + BC = AC$.

Use the distance formula to find the lengths of the three sides. An isosceles triangle has two equal sides. An equilateral triangle has three equal sides. A right-angled triangle satisfies the Pythagorean theorem, $a^2 + b^2 = c^2$.

To identify a quadrilateral, calculate the lengths of all four sides and both diagonals. A square has all sides equal and diagonals equal. A rhombus has all sides equal but unequal diagonals. A rectangle has opposite sides equal and diagonals equal.

Any point on the x-axis has coordinates of the form $(x, 0)$, as its distance from the x-axis (y-coordinate) is zero. Similarly, any point on the y-axis has coordinates of the form $(0, y)$.

To find the ratio in which a point divides a line segment, assume the ratio is $k:1$. Use the section formula to express the coordinates in terms of $k$, then equate with the given coordinates of the point to solve for $k$.

The points of trisection divide a line segment into three equal parts. For a segment AB, the first point divides it in the ratio $1:2$ and the second point divides it in the ratio $2:1$. Use the section formula for each case.

To find a point $P(x, y)$ equidistant from two other points $A$ and $B$, set the distances equal: $PA = PB$. It is often easier to work with the squared distances, $PA^2 = PB^2$, to avoid square roots.

The diagonals of a parallelogram bisect each other. This means the mid-point of one diagonal is the same as the mid-point of the other diagonal. This property is useful for finding a missing vertex.