A Square within a Rectangle. Construct a rectangle of sides 8 cm and 4 cm. How will you construct a square inside, as shown in the figure, such that the centre of the square is the same as the centre of the rectangle? (Hint: Draw a rough figure. What will be the sidelength of the square? What will be the distance between the corners of the square and the outer rectangle?)
Answer
Plan first. In the figure the square touches the top and the bottom sides of the rectangle. So its side must equal the height of the rectangle.
side of the square = height of rectangle = 4 cm
space left over along the length = 8 − 4 = 4 cm
this is shared equally on the two sides: 4 ÷ 2 = 2 cm on the left and 2 cm on the right
Steps of construction
Construct the rectangle ABCD with AB = 8 cm and BC = 4 cm (base, two perpendiculars, then the top side).
On the base AB mark a point 2 cm from A and another point 2 cm from B.
At each of these two points draw a perpendicular to AB, right across to the top side.
The strip between these two perpendiculars is the required square, 4 cm by 4 cm.
The 4 cm square sits in the middle of the 8 cm × 4 cm rectangle, leaving 2 cm on the left and 2 cm on the right. The green dot is the common centre.
Why 2 cm on each side: “same centre” means the picture must look the same from the left and from the right. The leftover length 8 − 4 = 4 cm therefore has to be split into two equal halves of 2 cm.
Q2.
Falling Squares. Construct three squares of side 4 cm arranged as shown (make sure that the squares are aligned the way they are shown). Now, try this with a square of side 3 cm, a square of side 5 cm and a square of side 7 cm.
Answer
Look carefully at how the squares are joined: the bottom-left corner of one square is the top-right corner of the next. They only touch, corner to corner, going down to the left.
Steps of construction (three squares of side 4 cm)
Construct the first square ABCD of side 4 cm at the top right (base, perpendiculars, top).
Take its bottom-left corner. Starting there, construct the second square of side 4 cm going down and to the left, so that this corner becomes the second square's top-right corner.
Repeat once more from the bottom-left corner of the second square to get the third square.
Keep every side either horizontal or vertical, so that the squares stay aligned like steps of a staircase.
Falling squares. Each red dot is a corner shared by two squares — the lower square hangs from the bottom-left corner of the one above it.
For the second figure the same rule is used, but the squares get bigger as they fall: side 3 cm at the top, then 5 cm, then 7 cm. Start with the 3 cm square; from its bottom-left corner build the 5 cm square (that corner is the 5 cm square's top-right corner); from the bottom-left corner of the 5 cm square build the 7 cm square.
Tip: draw one long horizontal helper line for each row of squares. If a square slips even a little, the “falling” look is lost.
Q3.
Shadings. Construct this. Choose measurements of your choice. Note that the larger 4-sided figure is a square and so are the smaller ones.
Answer
Steps of construction (taking a big square of side 8 cm)
Construct a square of side 8 cm.
Mark points every 2 cm along all four sides and join opposite marks. The square is now a grid of 16 small squares, each 2 cm × 2 cm.
In the small squares you want shaded, draw the diagonal from the top-left corner to the bottom-right corner and shade the triangle above it with slanting lines.
Leave a 4 cm square (four small squares together) unshaded at the bottom-left, and one more 2 cm square unshaded next to it, exactly as in the picture.
The shading pattern: each shaded cell is cut by a diagonal and the upper triangle is filled in. A 4 cm square and one 2 cm square at the bottom left are left plain.
Tip: all the diagonals slant the same way, so the shaded triangles line up and the whole design looks like falling rain. Shade lightly with a pencil, using strokes parallel to the diagonal.
Q4.
Square with a Hole. Observe that the circular hole is the same as the centre of the square. (Hint: Think where the centre of the circle should be.)
Answer
The centre of the circle must be the centre of the square, and the centre of a square is the point where its two diagonals cross.
Steps of construction (square of side 6 cm, hole of radius 1.5 cm)
Construct a square ABCD of side 6 cm.
Join the diagonals AC and BD with light pencil lines. Call their meeting point O.
Place the compass tip at O, open it to 1.5 cm and draw the circle.
Rub out the two diagonals — they were only helpers.
The diagonals of the square meet at O, its centre. The hole is the circle drawn with O as centre.
Why the diagonals give the centre: each diagonal cuts the square into two equal halves, so the mid-point of a diagonal is equally far from all four sides. Both diagonals pass through that one point, so their crossing is the centre — no measuring needed.
Q5.
Square with more Holes. Construct this.
Answer
This is the previous figure done four times. Divide the big square into four equal squares and put one hole in each.
Steps of construction (big square of side 8 cm)
Construct a square of side 8 cm.
Mark the mid-points of all four sides. Join the mid-point of the top side to the mid-point of the bottom side, and the mid-point of the left side to the mid-point of the right side. The big square is now four small squares of side 4 cm.
In each small square draw its two diagonals lightly to find its centre.
With each centre as the tip position, draw a circle of radius 1.5 cm. Use the same opening for all four, so that the holes are identical.
Rub out the light diagonals.
Four identical holes, one at the centre of each of the four small squares.
Tip: never change the compass opening between the four circles. If you re-set it each time, the holes come out slightly different and the design loses its symmetry.
Q6.
Square with Curves. This is a square with 8 cm sidelengths. (Hint: Think where the tip of the compass can be placed to get all the 4 arcs to bulge uniformly from each of the sides. Try it out!)
Answer
Each side of the square is replaced by an arc that bulges inwards. An arc always bends away from the compass tip, so for each side the tip must be placed outside the square, on the perpendicular bisector of that side.
A neat choice is a radius of 5 cm with the tip 3 cm outside the middle of the side, because 3, 4 and 5 fit together perfectly:
half of a side = 8 ÷ 2 = 4 cm
tip is 3 cm outside the middle of the side
distance from the tip to each corner = 5 cm (a 3–4–5 right triangle)
so an arc of radius 5 cm passes exactly through both corners
Steps of construction
Construct a square of side 8 cm.
Mark the mid-point of each side and draw a short perpendicular there, going outwards.
On each of these, mark the point 3 cm outside the square.
Open the compass to 5 cm. Put the tip on one of these four points and draw the arc joining the two corners of that side.
Repeat for the other three sides without changing the opening. All four arcs then bulge in by the same amount (2 cm).
Square with curves. For the bottom side the tip is 3 cm below the middle of the side; with radius 5 cm the arc passes through both corners and bulges 2 cm into the square. The other three sides are done in the same way.
Why all four arcs match: the four sides of a square are equal, so if the tip is put the same distance outside each side and the same radius is used every time, the four arcs must be identical. Using a larger radius makes the curves flatter; a smaller one (but always more than 4 cm) makes them deeper.