Practice Questions

Propose a method using the distance formula to determine if a point $P(x, y)$ lies inside, outside, or on a circle with center $C(h, k)$ and radius $r$.

Identify the quadrant in which the following points lie and state the sign of their abscissa and ordinate: (a) $(2, 5)$, (b) $(-3, 1)$, (c) $(-4, -6)$.

State the formula to find the distance between two points $P(x_1, y_1)$ and $Q(x_2, y_2)$.

Evaluate the statement: "The midpoint formula is a special case of the section formula." Justify your reasoning.

Calculate the distance of the point C(-8, 15) from the origin O(0, 0).

Calculate the distance between the points A(2, -3) and B(10, -9).

Define the term 'ordinate' of a point.

Solve for the coordinates of the point which divides the line segment joining A(-1, 7) and B(4, -3) in the ratio 2:3.

What are the coordinates of a point on the x-axis? Explain why the y-coordinate is zero for any point on the x-axis.

Write the coordinates of a point that lies on the y-axis at a distance of 5 units from the origin in the positive direction.

State the mid-point formula for a line segment joining the points $A(x_1, y_1)$ and $B(x_2, y_2)$.

Create a problem where the coordinates of a point $P$ that divides a line segment $AB$ in a given ratio are known, and you need to find the coordinates of an endpoint $B$, given endpoint $A$. Solve the problem for $A = (1, 5)$, $P = (4, -1)$, and the ratio $AP:PB = 3:2$.

Explain the condition for three points $A, B,$ and $C$ to be collinear using the distance formula.

Determine the ratio in which the point P(2, y) divides the line segment joining the points A(-2, 2) and B(3, 7). Also, solve for the value of y.

Find the point on the y-axis which is equidistant from the points M(5, -2) and N(-3, 2).

The vertices of a triangle are A(1, k), B(4, -3) and C(-9, 7). If its centroid is G(1, 4), calculate the value of k.

If A(6, 1), B(8, 2), C(9, 4) and D(p, 3) are the vertices of a parallelogram taken in order, demonstrate the property of its diagonals to find the value of p.

State the formula for the distance of a point $P(x, y)$ from the origin $O(0, 0)$.

Derive the formula for the coordinates of the centroid of a triangle with vertices $A(x_1, y_1)$, $B(x_2, y_2)$, and $C(x_3, y_3)$. (Hint: The centroid divides each median in the ratio 2:1).

Formulate the linear equation that represents the locus of all points $(x, y)$ that are equidistant from the points $A(-1, 4)$ and $B(5, -2)$. Interpret the geometric meaning of this equation.

State the section formula and explain what each variable represents.

Describe how you would use the distance formula to determine if a triangle with given vertices $A, B,$ and $C$ is an isosceles triangle.

A point P lies on the x-axis. If its distance from the point A(0, -4) is 5 units, calculate the possible coordinates of P.

Explain how the mid-point formula is a special case of the section formula.

List the conditions that must be satisfied for four points $A, B, C,$ and $D$ to be the vertices of a rhombus, using the distance formula.

Summarize the steps to prove that four given points $A, B, C,$ and $D$ form a parallelogram but not a rectangle, using the distance formula.

Describe the derivation of the distance formula, $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$, for two points $P(x_1, y_1)$ and $Q(x_2, y_2)$. Use a brief explanation of the geometric construction.

Explain the concept of 'trisection' of a line segment. Describe how to find the coordinates of the points of trisection for a line segment joining $A(x_1, y_1)$ and $B(x_2, y_2)$ using the section formula.

Examine if the points A(1, 1), B(3, 2), and C(7, 4) are collinear.

Analyze if the points P(3, 4), Q(-2, 3), and R(5, -6) form an isosceles triangle.

Show that the points A(-2, -1), B(1, 0), C(4, 3), and D(1, 2) are the vertices of a parallelogram. Is it a rectangle? Examine and justify.

Formulate the conditions, using the distance formula, for four points $A, B, C, D$ taken in order, to be the vertices of a rhombus but not a square.

A student claims that for three distinct points A, B, and C, if they are collinear, then it must be that $AB + BC = AC$. Justify whether this claim is always true for any arrangement of the collinear points.

Propose a coordinate-based method and find the coordinates of the circumcenter of a right-angled triangle whose vertices are $O(0,0)$, $A(2a, 0)$, and $B(0, 2b)$.

A line segment joining $A(2, -2)$ and $B(11, 7)$ is divided by a point $P(x, 4)$. Justify the ratio in which $P$ divides the line segment $AB$, and then find the x-coordinate of $P$.

Prove, using coordinate geometry, that the diagonals of a rectangle are equal in length and bisect each other.

Critique the following reasoning: "To find a point on the y-axis equidistant from $A(x_1, y_1)$ and $B(x_2, y_2)$, I must find the midpoint of AB."

Let $P, Q, R, S$ be the midpoints of the sides $AB, BC, CD, DA$ respectively of an arbitrary quadrilateral $ABCD$. Justify, using coordinate geometry, that $PQRS$ is a parallelogram.

Solve for the coordinates of the points that trisect the line segment joining P(4, -1) and Q(-2, -3).

Create a proof using coordinate geometry to show that the medians to the equal sides of an isosceles triangle are equal in length. (Hint: Place the vertices strategically on the coordinate axes, for example, at $A(0, a)$, $B(-b, 0)$, and $C(b, 0)$).

Determine the ratio in which the line $x - 3y = 0$ divides the line segment joining the points A(-2, -5) and B(6, 3). Also, find the coordinates of the point of intersection.

The vertices of a triangle are A(5, 1), B(-3, -7), and C(7, -1). Calculate the length of the median through vertex A and find the coordinates of the centroid.

Evaluate the type of triangle formed by the vertices $A(a, a)$, $B(-a, -a)$, and $C(-a\sqrt{3}, a\sqrt{3})$ for $a \neq 0$. Justify your answer with calculations.

Design a solution for the following problem: Two ancient wells are located at positions $W_1(-4, 2)$ and $W_2(2, 8)$ on a village map. A straight irrigation canal is to be built such that every point on the canal is equidistant from both wells.

1. Formulate the linear equation of the path of the canal.
2. If the canal intersects a straight road represented by the y-axis, find the coordinates of the intersection point.
3. Justify why this canal is the perpendicular bisector of the line segment joining the two wells.