Question

Instructions:
(i) Attempt any three out of the following five questions. All questions have equal marks.
(ii) In the construction, given lines and required lines must be clearly shown.
(iii) Do not erase construction lines.
(iv) Do all the rough work on white foolscap paper and stitch it with your answer
-sheet.
(a) Construct a triangle ABC with a perimeter of 12 cm, with sides in the ratio of 3:4:5.
b) Draw an indirect tangent line on two equal circles of radii 3 cm each and whose centres are 7 cm apart.
c) Construct a parallelogram ABCD whose two adjacent sides AB and AD are 5 cm and 6 cm, respectively, and the angle between these two is 60°. Construct a rectangle of equal area to the given parallelogram.
d) Construct a hexagon with its side equal to 5 cm.
e) Construct a simple scale, in which 4 cm represents 1 metre distance. Find its representative fraction and show 2 metres and 5 decimetres distance on it.

Hint:

In triangles with sides in a specific ratio, the perimeter is distributed according to that ratio to find the actual side lengths.
The indirect tangent touches both circles externally without intersecting them. The construction requires drawing perpendicular bisectors and using symmetry.
To create a rectangle with equal area to a parallelogram, the base and height of the parallelogram must be matched by the length and breadth of the rectangle.
A regular hexagon can be inscribed in a circle, where each side is equal to the radius of the circle.
The representative fraction (RF) for the scale is the ratio of the distance on the drawing to the real distance. For this scale, RF = \( \frac{4}{100} \).

Answer 1

(a) Construct a triangle ABC with a perimeter of 12 cm, with sides in the ratio of 3:4:5.

Step 1: Understanding the Problem.
We are given the perimeter of the triangle as 12 cm, and the sides are in the ratio 3:4:5. First, we will find the actual lengths of the sides using the given perimeter.
Step 2: Find the Lengths of the Sides.
Let the sides of the triangle be 3x, 4x, and 5x. The sum of these sides is equal to the perimeter: \[ 3x + 4x + 5x = 12 \quad \Rightarrow \quad 12x = 12 \quad \Rightarrow \quad x = 1. \] Thus, the sides of the triangle are: \[ 3x = 3 \, \text{cm}, \, 4x = 4 \, \text{cm}, \, 5x = 5 \, \text{cm}. \]
Step 3: Construction of Triangle.

- Draw a base \( AB = 5 \, \text{cm} \).
- Using a compass, set the radius to 3 cm and draw an arc from point A.
- Set the radius to 4 cm and draw an arc from point B.
- Mark the intersection of these arcs as point C.
- Join \( AC \) and \( BC \) to complete the triangle.
Conclusion:
Thus, triangle ABC with sides 3 cm, 4 cm, and 5 cm, and a perimeter of 12 cm is constructed.
b) Draw an indirect tangent line on two equal circles of radii 3 cm each and whose centres are 7 cm apart.
Step 1: Understanding the Problem.
We are asked to draw an indirect tangent line between two equal circles, each with a radius of 3 cm. The distance between their centres is 7 cm. The indirect tangent will touch both circles but not intersect them.
Step 2: Construction.

- Draw two circles with a radius of 3 cm and centres 7 cm apart. Label the centres as \( O_1 \) and \( O_2 \).
- Join the centres \( O_1 \) and \( O_2 \) with a straight line.
- Find the midpoint of this line and draw a perpendicular line from the midpoint to the line joining \( O_1 \) and \( O_2 \).
- Now, draw the two tangent lines from the midpoint of the line joining the centres. These lines will be the indirect tangents to the two circles.
Conclusion:
The indirect tangent lines have been successfully constructed for the given circles.
c) Construct a parallelogram ABCD whose two adjacent sides AB and AD are 5 cm and 6 cm, respectively, and the angle between these two is 60°. Construct a rectangle of equal area to the given parallelogram.
Step 1: Understanding the Problem.
We need to construct a parallelogram with two sides measuring 5 cm and 6 cm, and the angle between them is 60°. Then, we will construct a rectangle with the same area as the parallelogram.
Step 2: Construction of Parallelogram.

- Draw a line segment \( AB = 5 \, \text{cm} \).
- At point A, construct an angle of 60° using a protractor.
- Draw line \( AD = 6 \, \text{cm} \) from point A, making an angle of 60° with \( AB \).
- From point D, draw a line parallel to \( AB \) and label the intersection as \( C \).
- Join \( B \) and \( C \) to complete the parallelogram.
Step 3: Area of the Parallelogram.
The area of the parallelogram is given by the formula: \[ \text{Area of parallelogram} = \text{Base} \times \text{Height} = 5 \, \text{cm} \times 6 \, \text{cm} \times \sin(60°) = 30 \times \frac{\sqrt{3}}{2} \approx 25.98 \, \text{cm}^2. \]
Step 4: Construction of Rectangle.
To construct a rectangle with the same area, use the formula for the area of a rectangle: \[ \text{Area of rectangle} = \text{Length} \times \text{Breadth}. \] Choose appropriate dimensions for the rectangle, such as 5 cm for length and approximately 5.2 cm for breadth, to make the area equal to the parallelogram’s area.
Conclusion:
Thus, we have successfully constructed both the parallelogram and the rectangle with equal areas.
d) Construct a hexagon with its side equal to 5 cm.
Step 1: Understanding the Problem.
We are asked to construct a regular hexagon where each side is 5 cm. A regular hexagon has six equal sides and internal angles of 120°.
Step 2: Construction of Hexagon.

- Start by drawing a circle with a radius of 5 cm.
- Mark six equidistant points along the circumference of the circle. These points will be the vertices of the hexagon.
- Connect these points sequentially to form the hexagon.
Conclusion:
The regular hexagon with sides of 5 cm has been successfully constructed.
e) Construct a simple scale, in which 4 cm represents 1 metre distance. Find its representative fraction and show 2 metres and 5 decimetres distance on it.
Step 1: Understanding the Problem.
We need to construct a simple scale where 4 cm represents 1 metre, and we also need to calculate the representative fraction (RF) and show specific distances on the scale.
Step 2: Construction of the Scale.

- Draw a line of 4 cm length. This line will represent 1 metre.
- Divide the line into smaller parts: each centimetre on the scale will represent 25 cm (since 1 metre = 100 cm, and 4 cm = 100 cm on the scale). Thus, each 1 cm on the scale will represent 25 cm.
- To show 2 metres and 5 decimetres, calculate the equivalent in scale units:
- 2 metres = 200 cm, which equals \( \frac{200}{25} = 8 \) cm on the scale.
- 5 decimetres = 50 cm, which equals \( \frac{50}{25} = 2 \) cm on the scale.
- Mark 8 cm and 2 cm on the scale.
Conclusion:
The simple scale is drawn, and the distances for 2 metres and 5 decimetres are marked.