Hard Level Worksheet

Very Short Answer Questions (1 Mark Each)

(1) Write the coordinates of the point lying on the negative y-axis at a distance of 8 units from the origin.

Perfect! On negative y-axis, x = 0 and y = -8 (8 units below origin).

(2) Give an example of a point in the fourth quadrant. 3,-2-3,-2

Excellent! Fourth quadrant has positive x and negative y coordinates.

(3) Write the coordinates of the reflection of the point (4, 5) in the x-axis.

Correct! Reflection in x-axis changes the sign of y-coordinate only.

(4) Write the coordinates of the reflection of the point (-3, -7) in the y-axis.

Great! Reflection in y-axis changes the sign of x-coordinate only.

(5) Write the coordinates of the point that is equidistant from the origin and lies in the second quadrant.

Any point like -a,aa,a where a > 0

Perfect! Points like (-3, 3), (-5, 5) are equidistant from origin.

Short Answer Questions (2 Marks Each)

(1) If point P(a, 0) lies on the x-axis and is equidistant from Q(3, 0) and R(-3, 0), find a. a =

Excellent! The point P(0, 0) is equidistant from Q(3, 0) and R(-3, 0).

(2) The coordinates of one endpoint of a line segment are (2, 3) and the midpoint is (0, 1). Find the other endpoint. Other endpoint:

Perfect! The other endpoint is (-2, -1).

(3) Find the midpoint of the line segment joining A(-6, 7) and B(4, -3). Midpoint:

Excellent calculation!

(4) If the point P(2, y) lies on the x-axis, find y. y =

Correct! Points on x-axis have y = 0.

(5) A point is reflected in the y-axis. If the image is (-4, 7), find the original point. Original point is

Perfect! Reflecting (4, 7) in y-axis gives (-4, 7).

Long Answer Questions (4 Marks Each)

(1) Plot the points A(2, 3), B(-2, 3), C(-2, -3), and D(2, -3) on a graph paper. Join them in order and name the quadrilateral. Find the lengths of its diagonals.

Point A(2, 3): Quadrant IIIIIIIV

Point B(-2, 3): Quadrant IIIIIIIV

Point C(-2, -3): Quadrant IIIIIIIV

Point D(2, -3): Quadrant IVIIIIII

Quadrilateral:

Diagonal AC: 213210

Diagonal BD: 213313

Excellent! Both diagonals are equal, confirming it's a rectangle.

(2) The vertices of a triangle are A(1, 1), B(7, 3), and C(5, 7). Plot the points and find the length of each side using the distance formula.

Side AB: 210213

Side BC: 2552

Side CA: 213210

Perfect! It's a triangle.

(3) On a graph paper, plot the points P(0, 0), Q(0, 5), R(5, 5), and S(5, 0). Find the area of the figure formed by joining them in order.

Figure formed:

Side length: units

Area: square units

Excellent! The vertices form a square with area 25 square units.

(4) Plot the points M(-4, 0), N(0, 3), and O(4, 0). Find the perimeter of the triangle formed using the distance formula. Perimeter: units

Great! The triangle is isosceles with perimeter 18 units.

(5) The vertices of a quadrilateral are A(-3, -2), B(5, -2), C(5, 3), and D(-3, 3). Plot the points and find the perimeter of the quadrilateral. Perimeter: units

Perfect! It's a rectangle with perimeter 26 units.

Part B: Objective Questions (1 Mark Each)

Choose the correct answer and write the option (a/b/c/d)

(1) The reflection of (2, 5) in the x-axis is:

(a) (2, -5) (b) (-2, 5) (c) (-2, -5) (d) (5, 2)

Correct! Reflection in x-axis changes the sign of y-coordinate only.

(2) The reflection of (-4, 7) in the y-axis is:

(a) (4, 7) (b) (-4, -7) (c) (4, -7) (d) (-7, 4)

Correct! Reflection in y-axis changes the sign of x-coordinate only.

(3) The midpoint of the segment joining (2, 3) and (4, 7) is:

(a) (3, 5) (b) (6, 10) (c) (2, 7) (d) (4, 5)

Correct! Midpoint = (2+42, 3+72) = (3, 5).

(4) The length of the segment joining (0, 0) and (3, 4) is:

(a) 5 (b) 7 (c) 6 (d) 4

Correct! Distance = 32+42 = 9+16 = 25 = 5.

(5) The coordinates of a point lying in the third quadrant are:

(a) (-2, -3) (b) (2, -3) (c) (-2, 3) (d) (2, 3)

Correct! Third quadrant has both coordinates negative.

(6) If the midpoint of (x, 4) and (6, -2) is (5, 1), then x is:

(a) 2 (b) 4 (c) 6 (d) 8

Correct! Using midpoint formula: x+62 = 5, so x = 4.

(7) The point (0, 0) is equidistant from:

(a) All points on the plane (b) All points on the x-axis (c) All points on the y-axis (d) All four quadrants

Correct! The origin is equidistant from all four quadrants (distance = 0).

(8) Which formula is used to find the length of a line segment between (x1, y1) and (x2, y2)?

(a) x2 - x1 + y2 - y1 (b) x2−x12+y2−y12 (c) x1 + x2, y1 + y2 (d) None of these

Correct! This is the distance formula derived from Pythagorean theorem.

(9) If a point is reflected in both the x-axis and y-axis, (a, b) becomes:

(a) (a, -b) (b) (-a, b) (c) (-a, -b) (d) (b, a)

Correct! Both coordinates change sign when reflected in both axes.

(10) The coordinates of the point lying exactly midway between (2, 3) and (-2, -3) are:

(a) (0, 0) (b) (2, -3) (c) (-2, 3) (d) (1, 1)

Correct! Midpoint = (2+−22, 3+−32) = (0, 0).

Let's practice coordinate transformations!!!

Excellent! You understand all types of coordinate reflections.

True or False: Distance and Midpoint Properties

Determine whether these statements are True or False:

Comprehensive Hard Quiz

🎉 You Did It! What You've Mastered:

By completing this advanced worksheet, you now have expert knowledge of:

(1) Advanced Reflections: Understanding transformations across x-axis, y-axis, and both axes

(2) Distance Formula Mastery: Calculating exact distances between any two points using x2−x12+y2−y12

(3) Midpoint Formula Applications: Finding centers, bisectors, and equidistant points

(4) Complex Geometric Construction: Creating and analyzing rectangles, triangles, and squares through coordinates

(5) Perimeter and Area Calculations: Using coordinate geometry for practical measurements

(6) Transformation Properties: Understanding how reflections change coordinate signs

(7) Advanced Problem Solving: Applying multiple concepts to solve complex geometric problems

(8) Real-World Applications: Using coordinates for city planning, navigation, and spatial analysis

(9) Pattern Recognition: Identifying mathematical relationships in coordinate sequences

(10) Geometric Proofs: Using coordinate methods to prove geometric theorems

(11) Triangle Properties: Analyzing scalene, isosceles, and equilateral triangles through coordinates

(12) Quadrilateral Analysis: Understanding rectangles, squares, and other shapes via coordinate properties

(13) Advanced Calculations: Working with square roots, fractions, and exact values

(14) Verification Techniques: Checking answers through multiple coordinate geometry methods

Exceptional mastery of advanced coordinate geometry achieved!