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 common direct tangent on two unequal circles of radius 3 cm and 4 cm respectively whose centres are 8 cm apart.
(b) Construct an isosceles triangle ABC, Base AB = 7 cm and its height is 5 cm.
(c) In a semi
-circle of diameter 10 cm, draw a beautiful geometrical design.
(d) Construct a hexagon of side 6 cm each.
(e) Construct a circumcircle of triangle ABC whose sides are AB = 7 cm, BC = 6 cm, and AC = 5 cm.

Hint:

The common tangent is the straight line that touches both circles externally without crossing them.
In an isosceles triangle, the height from the apex to the base bisects the base into two equal parts.
In geometric designs, symmetry and evenly spaced elements enhance visual appeal.
To construct a regular hexagon, divide the circle into six equal parts, each with a 60° angle.
The circumcircle of a triangle is constructed by finding the perpendicular bisectors of the sides and their point of intersection (the circumcenter).

Answer 1

(a) Construct a common direct tangent on two unequal circles of radius 3 cm and 4 cm respectively whose centres are 8 cm apart.
Steps of Construction:
Step 1: Draw the Circles. Begin by drawing two circles with radii 3 cm and 4 cm. Mark the centres of the circles as \( O_1 \) and \( O_2 \), respectively. The distance between the centres \( O_1 \) and \( O_2 \) is given as 8 cm.
Step 2: Join the Centres. Join the centres \( O_1 \) and \( O_2 \) with a straight line. The length of this line is 8 cm.
Step 3: Construct the Direct Tangent. Using the formula for the length of a common tangent between two circles, draw a line that is the common direct tangent. The length of the direct tangent can be calculated using the formula: \[ L = \sqrt{d^2
- (r_1
- r_2)^2} \] where \( d \) is the distance between the centres, and \( r_1 \) and \( r_2 \) are the radii of the circles. In this case, \( d = 8 \), \( r_1 = 4 \), and \( r_2 = 3 \). Solve for the length of the tangent.
Step 4: Draw the Tangent. Draw the common tangent line, ensuring it touches both circles without intersecting them. Label the points where the tangent touches the circles as \( P_1 \) and \( P_2 \).

Concepts to Keep in Mind:
- The direct tangent to two unequal circles touches both circles externally at one point each.
- The formula for the length of the tangent line between two circles is crucial to solve this construction.
(b) Construct an isosceles triangle ABC, Base AB = 7 cm and its height is 5 cm.
Steps of Construction:
Step 1: Draw the Base. Start by drawing a base \( AB \) of length 7 cm.
Step 2: Mark the Midpoint. Mark the midpoint \( M \) of the base \( AB \).
Step 3: Draw the Height. From point \( M \), draw a perpendicular line upwards. This line represents the height of the triangle, and it should be 5 cm long.
Step 4: Complete the Triangle. From the top of the height line, use a compass to draw arcs from points \( A \) and \( B \), each with a radius equal to the length of the sides of the isosceles triangle. These arcs will meet at point \( C \), completing the triangle.
Step 5: Finalize the Construction. Connect points \( A \), \( B \), and \( C \) to complete the isosceles triangle.

Concepts to Keep in Mind:
- An isosceles triangle has two equal sides. The height divides the base into two equal parts.
- The perpendicular height is crucial for constructing the triangle correctly.
(c) In a semi
-circle of diameter 10 cm, draw a beautiful geometrical design.
Steps of Construction:
Step 1: Draw the Semi
-Circle. Start by drawing a diameter of 10 cm. Use this as the base of your semi
-circle.
Step 2: Draw the Arc. Use a compass with a radius of 5 cm (half of the diameter) to draw the arc that forms the semi
-circle above the base.
Step 3: Add Geometrical Design. Begin adding geometric shapes inside the semi
-circle. You can draw concentric circles, triangles, or other regular polygons within the semi
-circle. Experiment with different sizes and placements to create a harmonious design.
Step 4: Enhance the Design. Use lines and curves to link the geometric shapes, creating a balanced and attractive design. Ensure the shapes are evenly spaced, and add intricate details for aesthetic appeal.
Step 5: Colouring and Final Touches. Colour the design using a selection of vibrant colours, ensuring they complement each other and create an appealing visual effect. Pay attention to the placement of colours to maintain harmony.

Concepts to Keep in Mind:
- A semi
-circle provides a natural shape for creating intricate, circular designs.
- Symmetry and balance are key elements in creating a harmonious design.
(d) Construct a hexagon of side 6 cm each.
Steps of Construction:
Step 1: Draw a Circle. Start by drawing a circle with a radius of 6 cm using a compass.
Step 2: Mark the First Point. Mark a point on the circumference of the circle. This will be your first vertex.
Step 3: Construct the Hexagon. Using a protractor or compass, divide the circumference of the circle into 6 equal parts. Each part will be 60°. Mark these points on the circle.
Step 4: Connect the Points. Connect the consecutive points on the circumference of the circle with straight lines to form the hexagon.
Step 5: Finalize the Construction. Once the hexagon is constructed, you can shade or highlight the hexagon to enhance its appearance.

Concepts to Keep in Mind:
- A hexagon has six equal sides and angles of 120°.
- A circle is the best way to construct a regular hexagon, as its vertices must lie on the circumference.
(e) Construct a circumcircle of triangle ABC whose sides are AB = 7 cm, BC = 6 cm, and AC = 5 cm.
Steps of Construction:
Step 1: Draw the Triangle. Start by drawing triangle ABC with the given side lengths: \( AB = 7 \, \text{cm} \), \( BC = 6 \, \text{cm} \), and \( AC = 5 \, \text{cm} \).
Step 2: Perpendicular Bisectors. To construct the circumcircle, first draw the perpendicular bisectors of at least two sides of the triangle. Use a compass to find the midpoints of sides \( AB \) and \( AC \), then construct the perpendicular bisectors through these midpoints.
Step 3: Locate the Circumcenter. The point where the two perpendicular bisectors intersect is the circumcenter, which is the center of the circumcircle. Mark this point as \( O \).
Step 4: Draw the Circumcircle. Using a compass with \( O \) as the center, draw a circle passing through any of the vertices (for example, point \( A \)). This is the circumcircle of triangle ABC.
Step 5: Finalize the Construction. The circle should pass through all three vertices of the triangle. Ensure that all points are correctly placed and that the circle is drawn smoothly.

Concepts to Keep in Mind:
- The circumcenter is equidistant from all three vertices of the triangle.
- The circumcircle passes through all three vertices of the triangle.