Question

Instructions:
Attempt any three questions among the following five questions.
All questions carry equal marks.
In each answer given lines, construction lines and required lines must be clearly shown. Do not erase the construction lines.
Use white foolscap paper for rough work and attach it also with your answer sheet.
a) Construct a triangle ABC of which perimeter is 12 cm and its sides are in the ratio of 3:4:5.
b) Construct a triangle whose sides are 4 cm, 5 cm, and 6 cm. Construct a rectangle which is equal in area to this triangle.
c) Make a geometrical design in a square of 6 cm side. The design should be on the basis of triangle, square, and lines.
d) Construct a transverse tangent on two equal circles A and B of radii 3 cm each and the distance between their centres is 7 cm.
e) Construct a rectangle ABCD of area 15 sq. cm. Construct a square equal in area to this rectangle.

Hint:

Use accurate measurements for both the rectangle and the square, and ensure the area of the square equals the area of the rectangle.

Answer 1

a) Construct a triangle ABC of which perimeter is 12 cm and its sides are in the ratio of 3:4:5.
Steps of Construction: Step 1: Calculate the Lengths of the Sides.
- The sides of the triangle are in the ratio of 3:4:5, and the perimeter is 12 cm.
- Let the sides be \( 3x, 4x, 5x \), where \( x \) is a constant multiplier.
- The perimeter is the sum of the sides: \[ 3x + 4x + 5x = 12 \, \text{cm} \] \[ 12x = 12 \quad \Rightarrow \quad x = 1 \, \text{cm} \] Thus, the sides of the triangle are 3 cm, 4 cm, and 5 cm. Step 2: Drawing the Triangle.
- Draw a base \( AB = 4 \, \text{cm} \) using a ruler.
- From point \( A \), use a compass to mark a point \( C \) such that \( AC = 3 \, \text{cm} \).
- Similarly, from point \( B \), mark a point \( C \) such that \( BC = 5 \, \text{cm} \).
- The intersection of these two arcs gives the position of point \( C \). Draw line segments \( AC \) and \( BC \) to complete the triangle.
Concept to Keep in Mind:
- The triangle's sides follow the Pythagorean theorem, as 3:4:5 is a Pythagorean triple (right-angled triangle).
b) Construct a triangle whose sides are 4 cm, 5 cm, and 6 cm. Construct a rectangle which is equal in area to this triangle.
Steps of Construction:
Step 1: Constructing the Triangle.
- Draw a base \( AB = 6 \, \text{cm} \).
- Using a compass, mark a point \( C \) such that \( AC = 4 \, \text{cm} \) and \( BC = 5 \, \text{cm} \).
- The intersection of these two arcs gives the position of point \( C \). Draw line segments \( AC \) and \( BC \) to complete the triangle.
Step 2: Finding the Area of the Triangle.
- The area \( A \) of a triangle is given by: \[ A = \frac{1}{2} \times \text{base} \times \text{height} \] - Using Heron's formula to calculate the area for the given sides, we find that the area of the triangle is 9.6 \( \text{cm}^2 \).
Step 3: Constructing the Rectangle.
- Construct a rectangle with an area of 9.6 \( \text{cm}^2 \).
- Let the length of the rectangle be 6 cm (same as the base of the triangle).
- The width \( w \) can be found using the formula for the area of a rectangle: \[ \text{Area} = \text{length} \times \text{width} \quad \Rightarrow \quad w = \frac{9.6}{6} = 1.6 \, \text{cm} \] - Construct a rectangle with length 6 cm and width 1.6 cm.
Concept to Keep in Mind:
- The area of the rectangle should be equal to the area of the triangle, 9.6 \( \text{cm}^2 \).
c) Make a geometrical design in a square of 6 cm side. The design should be on the basis of triangle, square, and lines.
Steps of Construction:
Step 1: Draw the Square.
- Draw a square with a side length of 6 cm using a ruler and protractor. Step 2: Drawing the Triangle.
- Inside the square, draw an equilateral triangle by marking all three sides equal and aligning them centrally within the square. Use a compass to ensure each side of the triangle is 6 cm.
Step 3: Adding Lines.
- Draw lines connecting the midpoints of each side of the square to the vertices of the triangle, creating interesting angles and patterns inside the square.
Step 4: Detailing the Design.
- Add additional lines or geometric shapes such as smaller squares or triangles within the larger design to enhance the pattern.
Concept to Keep in Mind: - The design must focus on symmetry and harmony between the triangle, square, and lines.
d) Construct a transverse tangent on two equal circles A and B of radii 3 cm each and the distance between their centres is 7 cm.
Steps of Construction:
Step 1: Draw the Two Circles.
- Draw two equal circles with radii of 3 cm each. Place their centres 7 cm apart.
Step 2: Constructing the Transverse Tangent.
- To construct the transverse tangent, first draw a line connecting the two centres of the circles. This line will be the distance between the centres.
- Using a compass and geometric methods, construct a tangent line that touches both circles externally without crossing the line connecting the centres. The tangent should meet both circles at one point.
Step 3: Finalizing the Construction.
- Label the points where the tangent touches the circles, and verify that the tangent is perpendicular to the line connecting the two centres at the points of contact.
Concept to Keep in Mind: - A transverse tangent touches both circles externally, creating two points of contact, and the line between the centres is perpendicular to the tangent at these points.
e) Construct a rectangle ABCD of area 15 sq. cm. Construct a square equal in area to this rectangle.
Steps of Construction:
Step 1: Constructing the Rectangle.
- Construct a rectangle with an area of 15 \( \text{cm}^2 \). You can choose the dimensions of the rectangle such that the length and width multiply to give 15. For example, let the length be 5 cm and the width be 3 cm.
- Use a ruler and protractor to accurately construct the rectangle with the specified dimensions.
Step 2: Constructing the Square.
- The area of the square should be equal to 15 \( \text{cm}^2 \). To find the side length of the square, use the formula for the area of a square: \[ \text{Area} = \text{side}^2 \quad \Rightarrow \quad \text{side} = \sqrt{15} \approx 3.87 \, \text{cm} \] - Construct a square with side length 3.87 cm.
Concept to Keep in Mind:
- The area of the square must be equal to the area of the rectangle, 15 \( \text{cm}^2 \).