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Worksheet
Coordinate Geometry explores the relationship between algebra and geometry through the use of coordinate systems to represent geometric shapes and solve problems.
Coordinate Geometry - Practice Worksheet
Strengthen your foundation with key concepts and basic applications.
This worksheet covers essential long-answer questions to help you build confidence in Coordinate Geometry from Mathematics for Class X (Mathematics).
Questions
Explain the distance formula and derive it using the Pythagoras theorem.
Draw a diagram with points P and Q and a right-angled triangle to visualize the distances.
Find the coordinates of the point which divides the line segment joining the points (2, -3) and (5, 6) in the ratio 1:2.
Recall the section formula and substitute the given values carefully.
Prove that the points (1, 7), (4, 2), (-1, -1), and (-4, 4) are the vertices of a square.
Calculate all four sides and both diagonals to verify the properties of a square.
Find the area of the triangle formed by the points (0, 0), (4, 0), and (0, 3).
Plot the points to visualize the triangle and identify the base and height.
Determine if the points (1, 5), (2, 3), and (-2, -11) are collinear.
Use the area method or slope method to check collinearity.
Find the point on the x-axis which is equidistant from (2, -5) and (-2, 9).
Use the distance formula and set the distances equal to each other.
Find the coordinates of the centroid of the triangle whose vertices are (4, -1), (-2, -3), and (6, -5).
Recall the formula for the centroid of a triangle as the average of its vertices' coordinates.
Find the ratio in which the line segment joining the points (-3, 10) and (6, -8) is divided by (-1, 6).
Use the section formula and solve for the ratio k:1.
Find the area of the quadrilateral whose vertices, taken in order, are (-3, 2), (5, 4), (7, -6), and (-5, -4).
Divide the quadrilateral into two triangles and calculate their areas separately.
Find the coordinates of the point which is equidistant from the three points (3, 0), (0, 3), and (-3, 0).
Find the perpendicular bisectors of two sides of the triangle and their intersection point.
Coordinate Geometry - Mastery Worksheet
Advance your understanding through integrative and tricky questions.
This worksheet challenges you with deeper, multi-concept long-answer questions from Coordinate Geometry to prepare for higher-weightage questions in Class X Mathematics.
Questions
Find the distance between the points A(2, 3) and B(4, 1) using the distance formula. Also, verify your answer by plotting these points on a graph paper and measuring the distance between them.
Remember the distance formula is derived from the Pythagorean theorem. Plotting points accurately is key to verification.
Show that the points (1, 7), (4, 2), (-1, -1), and (-4, 4) form a square. Calculate the lengths of all sides and diagonals to support your answer.
A square has equal sides and equal diagonals. Use the distance formula for each side and diagonal.
Find the coordinates of the point which divides the line segment joining the points (4, -3) and (8, 5) in the ratio 3:1 internally.
The section formula for internal division is (m1x2 + m2x1)/(m1 + m2), (m1y2 + m2y1)/(m1 + m2).
Determine if the points (1, 5), (2, 3), and (-2, -11) are collinear using the distance formula.
For collinearity, the sum of distances between two pairs should equal the distance between the first and last point.
Find the ratio in which the y-axis divides the line segment joining the points (5, -6) and (-1, -4). Also, find the point of intersection.
For division by y-axis, set x-coordinate of the dividing point to 0 and solve for the ratio.
Calculate the area of the triangle formed by the points (3, 0), (4, 5), and (-1, 4) using the formula for area of a triangle given vertices.
Use the formula: 1/2 |x1(y2-y3) + x2(y3-y1) + x3(y1-y2)|.
Find the coordinates of the points of trisection of the line segment joining the points A(2, -2) and B(-7, 4).
Trisection points divide the segment into three equal parts, so use section formula for ratios 1:2 and 2:1.
Prove that the points A(6, 1), B(8, 2), C(9, 4), and D(p, 3) are the vertices of a parallelogram. Find the value of p.
Use the property that diagonals of a parallelogram bisect each other to find p.
Find the coordinates of a point A, where AB is the diameter of a circle whose center is (2, -3) and B is (1, 4).
The center of the circle is the midpoint of the diameter AB.
Find the area of a rhombus if its vertices are (3, 0), (4, 5), (-1, 4), and (-2, -1) taken in order.
Area of a rhombus is half the product of its diagonals. Calculate diagonals using distance formula between opposite vertices.
Coordinate Geometry - Challenge Worksheet
Push your limits with complex, exam-level long-form questions.
The final worksheet presents challenging long-answer questions that test your depth of understanding and exam-readiness for Coordinate Geometry in Class X.
Questions
Prove that the points (3, 0), (6, 4), and (-1, 3) form a right-angled triangle. Also, find the area of the triangle.
Calculate the distances AB, BC, and AC using the distance formula. Check if the sum of the squares of the two shorter sides equals the square of the longest side.
Find the coordinates of the point which divides the line segment joining the points (2, -2) and (-7, 4) in the ratio 2:1 externally. Discuss the significance of external division in coordinate geometry.
Remember that for external division, the ratio is taken as k:1 where k is greater than 1. Think about how this concept is used in physics or engineering.
A point P divides the line segment joining A(1, -5) and B(-4, 5) in the ratio k:1. Find the value of k if P lies on the x-axis. What does this imply about the position of P?
The y-coordinate of P is 0. Use this to set up an equation using the section formula.
Determine the ratio in which the line segment joining the points (-3, 10) and (6, -8) is divided by the point (-1, 6). Verify your answer using the distance formula.
Let the ratio be k:1. Use the section formula to find k. Then, calculate the distances PA and PB to verify.
If the points A(6, 1), B(8, 2), C(9, 4), and D(p, 3) are the vertices of a parallelogram, find the value of p. Explain the properties of a parallelogram used in this problem.
The mid-points of the diagonals AC and BD should be the same. Use this to find p.
Find the area of the quadrilateral formed by the points (3, 0), (4, 5), (-1, 4), and (-2, -1) taken in order. Discuss the method used and its limitations.
Use the shoelace formula for each triangle and add the areas. Consider how this method might not work for self-intersecting polygons.
A circle has its center at (2, -3) and one end of a diameter at (1, 4). Find the coordinates of the other end of the diameter. Explain the geometric principle used.
The mid-point of the diameter is the center. Use the mid-point formula in reverse to find the other end.
Find the coordinates of the point on the y-axis which is equidistant from the points (6, 5) and (-4, 3). Discuss the significance of equidistant points in coordinate geometry.
The point is on the y-axis, so its x-coordinate is 0. Set the distances from (0, y) to both given points equal and solve for y.
Show that the points (1, 7), (4, 2), (-1, -1), and (-4, 4) are the vertices of a square. Verify using both distance and slope formulas.
All sides of a square are equal, and the diagonals are equal. Adjacent sides are perpendicular, so their slopes are negative reciprocals.
Find the relation between x and y such that the point (x, y) is equidistant from the points (7, 1) and (3, 5). Interpret this relation geometrically.
The set of points equidistant from two given points is the perpendicular bisector of the line segment joining them. Find the equation of this line.
Explore the world of Polynomials, understanding their types, degrees, and operations to solve algebraic expressions and equations effectively.
Explore the methods to solve a pair of linear equations in two variables, including graphical, substitution, elimination, and cross-multiplication techniques.
Explore the world of quadratic equations, learning to solve them using various methods like factoring, completing the square, and the quadratic formula.
A chapter that explores sequences where each term after the first is obtained by adding a constant difference, focusing on their properties, nth term, and sum formulas.
Explore the properties, types, and theorems related to triangles, including congruence and similarity, to solve geometric problems effectively.
