1. Geometry and Friction
1. (a) Define Polygon:
A polygon is a flat, two-dimensional shape with straight sides that are fully closed. The sides are called edges, and the points where they meet are called vertices.
(b) Construct the following polygons:
- Pentagon
- Hexagon
- Octagon
2. (a) Define Friction:
Friction is the force that opposes the relative motion of two surfaces in contact.
(b) State two advantages and two disadvantages of friction:
Advantages:
- Provides Traction: Enables walking and vehicle movement without slipping.
- Helps in Stopping: Assists in slowing down or stopping moving objects.
Disadvantages:
- Causes Wear and Tear: Leads to the wear and damage of materials like tires and machine parts.
- Increases Energy Consumption: Requires more energy to overcome resistance in machines.
(c) Differentiate between Lubrication and Lubricants:
Lubrication:
The process of applying a substance to reduce friction between moving parts.
Lubricants:
Substances such as oils or greases used to reduce friction and wear between surfaces.
(d) Mention two lubricants used in reducing friction:
- Motor Oil
- Grease
3. (a) Explain five care and maintenance practices for woodworking machines:
- Regular Cleaning: Remove sawdust and debris to keep machines operating smoothly.
- Lubrication: Apply oil or grease to moving parts to reduce friction and wear.
- Check for Loose Parts: Inspect and tighten bolts and screws to ensure safety and accuracy.
- Sharpen Tools: Keep cutting tools sharp to improve efficiency and reduce strain on the machine.
- Inspect Electrical Components: Regularly check and maintain electrical parts to prevent malfunctions.
(b) Explain two functions of a lathe machine:
- Turning: Rotates a workpiece to shape it using cutting tools.
- Drilling: Allows for boring holes into a workpiece using a drill attachment.
4. Sketch five workshop hand tools and explain their uses:
- Hammer: Drives nails, breaks objects, and shapes materials.
- Screwdriver: Turns screws to fasten or unfasten them.
- Wrench: Grips and turns nuts and bolts.
- Pliers: Grips, twists, and cuts wires or small objects.
- Saw: Cuts wood or other materials into desired shapes.
5. (a) Explain the uses of the following technical drawing instruments:
- Tee Square: Draws horizontal lines and checks angles.
- Protractor: Measures and draws angles.
- Set Square: Draws right angles and other specific angles.
- Compass: Draws circles and arcs.
- Divider: Measures and transfers distances.
(b) Sketch a tee square and label its parts:
- Draw a right-angle L-shape.
- Label the Blade (long part) and Head (short part perpendicular to the blade).
6. Geometry Problems
1. Construct a rectangle ABDE equal in area to a given triangle with sides AB = 70mm, AC = 80mm, and BC = 80mm. Prove that the area of rectangle ABDE equals the area of triangle ABC.
Solution:
To find the area of the triangle ABC, use Heron’s formula. First, find the semi-perimeter:
- Semi-perimeter (s) = (AB + AC + BC) / 2 = (70 + 80 + 80) / 2 = 115 mm
- Area (A) = sqrt(s * (s – AB) * (s – AC) * (s – BC))
Area (A) = sqrt(115 * (115 – 70) * (115 – 80) * (115 – 80))
Area (A) ≈ 2800 mm²
To find a rectangle with this area, if one side is 70mm, the other side will be:
- Width = 2800 mm² / 70 mm = 40 mm
Thus, a rectangle ABDE with length 70mm and width 40mm will have the same area as the triangle ABC.
2. Inscribe a circle into an isosceles triangle with sides AB = 80mm, AC = 80mm, and BC = 70mm.
Solution:
To inscribe a circle, use the formula for the radius (r) of the circle inscribed in a triangle:
- Radius (r) = Area / Semi-perimeter
- Semi-perimeter (s) = (AB + AC + BC) / 2 = (80 + 80 + 70) / 2 = 115 mm
- Area = sqrt(s * (s – AB) * (s – AC) * (s – BC))
Area ≈ 2470 mm²
- Radius (r) = 2470 mm² / 115 mm ≈ 21.5 mm
3. Inscribe a square into a circle with a radius of 30mm.
Solution:
The diagonal of the square is equal to the diameter of the circle. The diameter is:
- Diameter = 2 × Radius = 2 × 30 mm = 60 mm
Let the side of the square be s. The relationship between the diagonal and side of a square is:
- Diagonal = s√2
- s√2 = 60 mm
s = 60 mm / √2 ≈ 42.4 mm
4. Prove that triangles on the same base and between parallel lines have equal areas, given the base of the two triangles as 70mm and the distance between the two parallel lines as 45mm.
Solution:
Let the base of the triangles be b and the distance between the parallel lines be h. The area of each triangle is:
Since the base and height are the same for both triangles:
- Area1 = 1/2 × 70 mm × 45 mm
- Area2 = 1/2 × 70 mm × 45 mm
Thus, both triangles have equal areas.
