MATHS 5 Marks Questions

First draw a square of side 7 cm, named as ABCD.

Now, using a compass draw arcs above the side AB of radius 7 cm from points A and B. Let the arcs intersect at point E.

Joint A to E and B to E by straight lines

Now, take 7 cm radius in the compass and from E, draw the arc touching A and B as shown in the figure.

First, draw a rough diagram

Step 1: The base CD measuring length 4 cm can be easily constructed.

Step 2: Draw a perpendicular to line DC at the point C. Let us call this line l.

Step 3: Draw an arc of radius 8 cm with point D as the center.

To locate the point B,
Step 4: Draw perpendiculars to DC and BC passing through D and B, respectively. The point where these lines intersect is the fourth point A.

It satisfies all conditions of the rectangle.

Draw a horizontal line segment AB. This will be one side of the rectangle.

(i) At point A, use a protractor to measure and draw AX a 45° angle.

(ii) At point B, measure and draw a 90° angle. Draw a line segment BC extending from B at this angle meeting AX at C.

(iii) At Point A and C, Draw a 90° angle which meets at the point D.

ABCD is the required rectangle. Here, diagonal AC divides the opposite angles A and C into 45° and 45°.
Observation You will notice that when the diagonals divide the opposite angles into 45° and 45°, so the sides of the rectangle will be equal, forming a square.

We shall draw a rectangle of the form shown in Fig. 1.

Step 1. Using a ruler, draw a line AB equal to 4 cm, say. (Fig. 2.)

Step 2. Using a protractor, mark dots C and D at angles 50° and 90° (50° + 40°), keeping the central point of the protractor at A. (Fig. 3)

Step 3. Using a protractor, draw a perpendicular line to AB at B and let it intersect the extended line AC at E. (Fig 3)
Step 4. Using a protractor, draw a perpendicular line to BE at E and let it intersect the extended line AD at F. (Fig 4)

Step 5. Erase the extra lines in Fig. 4. (Fig. 5)

Step 6. Fig. 5 is the required rectangle in which one of the diagonals divides the opposite angles into 50° and 40°.

In the given figure, the centres of the four arcs are outside the square.
Step 1. Using a ruler, draw a line AB equal to 8 cm. Using a protractor, draw perpendicular lines at A and B. Using a ruler, mark point P on the perpendicular line at A such that AP = 8 cm. Using a ruler, mark point Q on the perpendicular line at B such that BQ = 8 cm. Join P and Q using a ruler. Erase the lines above P and Q. (Fig. 1)

Step 2. Using a ruler, mark points C, D, E, and F such that AC = 4 cm, BD = 4 cm, QE = 4 cm, and PF = 4 cm. Join C and E and also D and F. Extend these lines outside the square. (Fig. 2)

Step 3. Extend DF and take points G and H on it so that DG and FH are equal to 4 cm. Extend CE and take points I and J on it so that Cl and EJ are equal to 4 cm. The distance 4 cm can be taken slightly less than or greater than 4 cm. Join B and G. (Fig. 3)

Step 4. With centres at G, H, I, and J and a radius equal to BG, draw four arcs inside the square as shown in the given figure. Erase the extra lines. (Fig. 4).

Step 5. Fig. 4 is the required “Square with Curves” with the square of side 8 cm.

In the figure, the centre of a circle is the same as that of the corresponding square.
Step 1. Using a ruler, draw a line AB equal to 8 cm, say. Using a protractor, draw perpendicular lines at A and B. Using a ruler, mark point P on the perpendicular line at A such that AP = 8 cm. Using a ruler, mark point Q on the perpendicular line at B such that BQ = 8 cm. Join P and Q using a ruler. Erase the lines above P and Q. (Fig. 1)

Step 2. Using a ruler, find points C, D, E, and F such that AC = 4 cm, BD = 4 cm, QE = 4 cm, and PF = 4 cm. Join C and E and also F and D. (Fig. 2)

Step 3. Let G be the intersection of lines FD and CE. Find the centres of squares ACGF, CBDG, DQEG, and GEPF by joining their respective diagonals. (Fig. 3)

Step 4. Erase the extra lines used for finding the centres of the smaller circles. With centre at centres of small squares, draw four circles of radius 1.3 cm, say. (Fig. 4)

Step 5. Fig. 4 is the required “‘Square with Four Holes”.

The centre of a square is the point of intersection of its diagonals. This centre is also the centre of the hole in the figure.
Step 1. Using a ruler, draw a line AB equal to 5 cm, say. Using a protractor, draw perpendicular lines at A and B. Using a ruler, mark point P on the perpendicular line at A such that AP = 5 cm. Using a ruler, mark point Q on the perpendicular line at B such that BQ = 5 cm. Join P and Q using a ruler. Erase the lines above P and Q (Fig. 1).

Step 2. Draw diagonals AQ and BP using a ruler. Let the diagonals intersect at C. This point is the centre of the square ABQP. Erase the diagonals AQ and BP. (Fig. 2).

Step 3. With centre at C and a radius of 1.5 cm, say, draw a circle using a compass. (Fig. 3)

Step 4. Fig. 3 is the required “Square with a Hole”.

Step 1. Using a ruler, draw a line AB equal to 8 cm. Because, 8 ÷ 4 = 2, we shall draw smaller squares of side 2 cm. Using a protractor, draw perpendicular lines at A and B. Using a ruler, mark point P on the perpendicular line at A such that AP = 8 cm. Using a ruler, mark point Q on the perpendicular line at B such that BQ = 8 cm. Join P and Q using a ruler. Erase the lines above P and Q (Fig. 1).

Step 2. On the lines AB, BQ, QP, and PA, mark points at distances of 2 cm, using a ruler. Draw horizontal lines and vertical lines to get 16 squares. (Fig. 2)

Step 3. From comer A, erase the inner sides of four squares to get a square of side 4 cm with one comer at A. Draw parallel diagonals of the remaining 12 small squares of side 2 cm each. (Fig. 3)

Step 4. In the 12 small squares, draw horizontal lines in the portion above the diagonals. (Fig. 4)

Step 5. Fig. 4 is the required figure having 12 small squares in a square.

In the given figure, there are three falling squares and the side of each square is 4 cm.
Step 1. Using a ruler, draw a line AB equal to 4 cm. Using a protractor, draw perpendicular lines at A and B.
Using a ruler, mark point C on a perpendicular line at A such that AC = 4 cm.
Using a ruler, mark points D and E on a perpendicular line at B such that BD = 4 cm and DE = 4 cm. (Fig. 1)

Step 2. Join C and D. Produce CD to F such that DF = 4 cm. Using a protractor, draw a perpendicular line at F. Using a ruler, mark points G and H on a perpendicular line at F such that FG = 4 cm and GH = 4 cm. (Fig. 2).

Step 3. Join E and G. Produce EG to I such that GE = 4 cm. Using a protractor, draw a perpendicular line at I. Using a ruler, mark point J on the perpendicular line at I such that IJ = 4 cm. Join H and J. Erase extra lines in the figure. (Fig. 3).

Step 4. Fig. 3 is the required figure of three “falling squares” each of side 4 cm.

Draw a rectangle PQRS of sides 8 cm and 4 cm.

Next, draw lines LN and MO from the midpoints of opposite sides such that they intersect side PQ at L, QR at M, RS at N, and PS at O respectively.

Since the centre of square and rectangle is same and the width of rectangle is 4 cm, so we draw a square of side 4 cm with the centre of rectangle and named as ABCD.

∴ The side length of the square is 4 cm and the distance between the comers of the square and the outer rectangle = PL – AL = 4 cm – 2 cm = 2 cm.

(i) Let the smaller side of a rectangle be x cm. If the larger side of the rectangle is 2x cm (x cm + x cm), then this rectangle can be divided into two identical squares of side x cm. (Fig. 1)

Let us consider a rectangle of sides 4 cm and 6 cm. Here, 6 is not equal to 8 (4 + 4), so, it cannot be divided into two identical squares as shown in Fig. 2

(ii) Let the smaller side of a rectangle be x cm. If the larger side of the rectangle is 3x cm (x cm + x cm + x cm), then this rectangle can be divided into three identical squares of side x cm. (Fig. 3)

Let us consider a rectangle of sides 3 cm and 8 cm. Here, 8 is not equal to 9 (3 + 3 + 3), so, it cannot be divided into three identical squares as shown in Fig. 4.

We shall draw a rectangle of the form shown in Fig. 1.

Step 1. Let us keep the vertical side of the rectangle to 3 cm. Since the rectangle is to be divided into three identical squares, the length of the rectangle must be 3 cm + 3 cm + 3 cm = 9 cm.
Step 2. Using a ruler, draw a line AB equal to 9 cm. (Fig. 2).

Step 3. Using a ruler, find points P and Q on AB such that AP = 3 cm and PQ = 3 cm. Here, QB is also 3 cm. (Fig. 3).

Step 4. Using a protractor, draw perpendicular lines at A, P, Q, and B. (Fig. 4).

Step 5. Using a ruler, mark points A’, P’, Q’, and B’ on perpendiculars at A, P, Q, and B respectively such that AA’ = PP’ = QQ’ BR’ = 3 cm. (Fig. 5).

Step 6. Join A’ and P’, P’ and Q’, and Q’ and B’ using a ruler. Erase the lines above A’, P’, Q’, and B’. (Fig. 6).

Step 7. ABB’A’ is the required rectangle which is divided into 3 identical squares APP’A’, PQQ’P’, and QBB’Q’.

Rectangle with sides 4 cm and 6 cm.
Steps
(i) Draw a,straight line AB = 6 cm using a ruler.

(ii) Place the protractor on point A and mark a 90° angle from AB.
(iii) Draw a line from A along this angle and measure AD 4 cm.

(iv) Repeat the same process at point B to draw a line BC perpendicular to AB and also 4 cm long.

(v) Join points D and C to complete the rectangle ABCD.

(vi) Verify that AB and CD are equal as well as AD and BC and all angles are 90°.