Question 2

A quadrilateral with equal diagonals that bisect each other is a rectangle. If these diagonals are also perpendicular, the quadrilateral is a square.

Step 1 — Draw the first diagonal

We need to draw a quadrilateral whose diagonals are 8 cm long. Let us start by drawing one of these diagonals. We draw a line segment and label its endpoints A and B. The length of this segment AB is 8 cm.

Step 2 — Locate the midpoint of the diagonal

The problem states that the diagonals bisect each other. This means they cut each other exactly in half at their intersection point. Let M be the midpoint of the diagonal AB. So, the distance from A to M is half of AB. The distance from B to M is also half of AB.

$AM = MB = \frac{\text{Length of diagonal}}{2}$

$AM = MB = \frac{8 \text{ cm}}{2}$

$\boxed{AM = MB = 4 \text{ cm}}$

We mark point M exactly in the middle of AB.

Step 3 — Draw the second diagonal at the specified angle

Now we need to draw the second diagonal. Let us call it CD. This diagonal must also be 8 cm long and pass through point M. It must make a specific angle with the first diagonal AB at point M. We use a protractor to draw a line through M. This line will form the second diagonal, CD. The angle it makes with MB will vary for each part of the question.

Step 4 — Mark the endpoints of the second diagonal

Since the second diagonal CD is also 8 cm long and is bisected by M, its halves must be 4 cm each. So, the distance from M to C is 4 cm. The distance from M to D is also 4 cm. We mark point C on the line drawn in Step 3, 4 cm away from M. We mark point D on the same line, on the opposite side of M, also 4 cm away from M.

$MC = MD = \frac{\text{Length of diagonal}}{2}$

$MC = MD = \frac{8 \text{ cm}}{2}$

$\boxed{MC = MD = 4 \text{ cm}}$

Step 5 — Complete the quadrilateral

Finally, we connect the endpoints of the two diagonals. We draw line segments connecting A to C, C to B, B to D, and D to A. This forms the required quadrilateral, ACBD.

Answer

(i) Draw a line AB equal to 8 cm. Take point M on AB such that AM = BM = 4 cm. Using a protractor, draw an angle of 30 degrees at M on MB. On this line, take points C and D such that MC = MD = 4 cm. Join AD, DB, BC, and CA. ABCD is the required quadrilateral. Since diagonals AB and CD are equal and are bisecting each other at M, ACBD is a rectangle. (ii) Draw a line AB equal to 8 cm. Take point M on AB such that AM = BM = 4 cm. Using a protractor, draw an angle of 40 degrees at M on MB. On this line, take points C and D such that MC = MD = 4 cm. Join AD, DB, BC, and CA. ABCD is the required quadrilateral. Since diagonals AB and CD are equal and are bisecting each other at M, ACBD is a rectangle. (iii) Draw a line AB equal to 8 cm. Take a point M on AB such that AM = BM = 4 cm. Using a protractor, draw an angle of 90 degrees at M on MB. On this line, take points C and D such that MC = MD = 4 cm. Join AD, DB, BC, and CA. ACBD is the required square. Since diagonals AB and CD are equal and are bisecting each other at M, and also the diagonals are perpendicular to each other, ACBD is a square. (iv) Draw a line AB equal to 8 cm. Take a point M on AB such that AM = BM = 4 cm. Using a protractor, draw an angle of 140 degrees at M on MB. On this line, take points C and D such that MC = MD = 4 cm. Join AD, DB, BC, and CA. ACBD is the required quadrilateral. Since diagonals AB and CD are equal and are bisecting each other at M, ACBD is a rectangle.