Question

(a) Construct a geometric plot of a hexagon of side 6 cm based on triangles.
(b) Construct a circle whose radius is 3.6 cm. Point B is 8 cm away from the center of the circle A. From point B, draw a tangent to the given circle.
(c) Construct a square equal to the area of an equilateral triangle of side 7 cm.
(d) Construct an equilateral triangle of 8 cm on the line AB.
(e) Construct a triangle equal to the area of a circle P.

Hint:

Use a compass and ruler for precision in geometric constructions. Ensure accurate measurements and check symmetry or area calculations to verify each shape.

Answer 1

Part (a): Hexagon based on triangles
Step 1 (a): Draw a horizontal line segment AB of 6 cm using a ruler. Using a compass set to 6 cm, draw arcs from A and B to intersect above and below the line, marking points C and D. This forms an equilateral triangle ABD. Repeat this process from points along AB (e.g., divide AB into three 2 cm segments and use points to form additional triangles), connecting all vertices to create a regular hexagon with side 6 cm, composed of six equilateral triangles. 
Part (b): Circle with tangent 
Step 1 (b): Draw a point A as the center. Set the compass to a radius of 3.6 cm and draw a circle around A. Mark point B 8 cm from A (using a ruler, ensuring AB > 3.6 cm, as 8 cm > radius 3.6 cm allows for an external tangent). With the compass at B and radius equal to the distance from B to the point of tangency (calculated as \(\sqrt{8^2 - 3.6^2} \approx 7.15 \ \text{cm}\)), draw an arc intersecting the circle. Draw a line from B through the intersection point tangent to the circle. 
Part (c): Square equal to triangle area 
Step 1 (c): Calculate the area of an equilateral triangle with side 7 cm: Area = \(\frac{\sqrt{3}}{4} \times 7^2 \approx 21.22 \, \text{cm}^2\). Find the side of a square with equal area: Side = \(\sqrt{21.22} \approx 4.61 \, \text{cm}\). Using a ruler and compass, construct a square with side 4.61 cm by drawing perpendicular lines and equal segments. 
Part (d): Equilateral triangle on line AB 
Step 1 (d): Draw line segment AB of 8 cm. Set the compass to 8 cm, place the pivot at A, and draw an arc. Place the pivot at B and draw another arc intersecting the first arc at C. Connect A to C and B to C to form an equilateral triangle ABC with side 8 cm.
Part (e): Triangle equal to circle area 
Step 1 (e): Assume circle P has a given radius (e.g., 3 cm for illustration). Calculate its area: Area = \(\pi \times 3^2 \approx 28.27 \, \text{cm}^2\). To construct a triangle with this area, use the formula for an equilateral triangle’s area: \(\frac{\sqrt{3}}{4} \times \text{side}^2 = 28.27\). Solve for side: \(\text{side}^2 \approx \frac{28.27 \times 4}{\sqrt{3}} \approx 65.26\), \(\text{side} \approx 8.08 \, \text{cm}\). Construct an equilateral triangle with side 8.08 cm using the method from (d).