MathSphere Free Sample Maths Worksheets - Free Printable

Let’s solve each problem step by step.

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Problem 1: Rectangle PQRS

We are given:
- P(1, 6)
- S(1, 2)
- R(8, 2)

Since it’s a rectangle, opposite sides are equal and parallel.

Looking at points P and S: same x-coordinate (x=1), so PS is vertical.
Points S and R: same y-coordinate (y=2), so SR is horizontal.

So, to find Q:
- It must be directly above R (same x as R → x=8)
- And directly right of P (same y as P → y=6)

→ So Q is at (8, 6)

Now, diagonals of a rectangle connect opposite corners: PR and QS.

The middle (midpoint) of the rectangle is where the diagonals cross — which is also the midpoint of either diagonal.

Let’s use diagonal from P(1,6) to R(8,2):

Midpoint formula:
x = (x₁ + x₂)/2 = (1 + 8)/2 = 9/2 = 4.5
y = (y₁ + y₂)/2 = (6 + 2)/2 = 8/2 = 4

→ Midpoint is (4.5, 4)

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Problem 2: Trapezium ABCD with reflective symmetry

Given:
- A(20, 35)
- D(12, 10)
- C(50, 10)
- Dotted line is axis of symmetry (vertical line through center)

Since it has reflective symmetry about the dotted line, point B should be the mirror image of point A across that line.

First, find the axis of symmetry.

Look at base DC: D(12,10) and C(50,10). The midpoint of DC is:

x = (12 + 50)/2 = 62/2 = 31

So the dotted line is x = 31.

Point A is at x=20. Distance from A to line x=31 is: 31 - 20 = 11 units.

So point B should be 11 units to the RIGHT of x=31 → 31 + 11 = 42

Same y as A → y=35

→ So B is at (42, 35)

Now, draw diagonals AC and BD.

We need to find where they meet — their intersection point.

But since the trapezium is symmetric, the diagonals will intersect ON the axis of symmetry → so x-coordinate of intersection is 31

We don’t even need to calculate y — question only asks for x-coordinate.

→ Answer: 31

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Problem 3: Rectangle with corners A(27,43), B(51,43), C(51,15), D(27,15)

Find the centre (midpoint of diagonals).

Use diagonal from A(27,43) to C(51,15):

Midpoint x = (27 + 51)/2 = 78/2 = 39
Midpoint y = (43 + 15)/2 = 58/2 = 29

→ Centre is (39, 29)

You can check with other diagonal B to D:
B(51,43), D(27,15)
x = (51+27)/2 = 78/2 = 39
y = (43+15)/2 = 58/2 = 29 → Same!

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Final Answer:

1. Q: (8, 6); Middle: (4.5, 4)
2. B: (42, 35); x-coordinate of diagonal meeting point: 31
3. Centre: (39, 29)