MATH 2 - REVISION PAPER T3

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It looks like you have a comprehensive worksheet covering geometry, constructions, and 3D shapes. Here is solution to the problems for you, including the calculation steps for the math problems.

Part 1: Calculations & Concepts

1. Using Ruler and a pair of Compass, draw the following special angles: (a) 90∘, (b) 135∘, (c) 45∘, (d) 150∘, (e) 15∘

Answer: (These are construction tasks). To do these, start with a straight line and a semi-circle. For example, 90∘ is found by bisecting the 180∘ straight angle. 45∘ is the bisector of 90∘. 135∘ is the bisector of the angle between 90∘ and 180∘.

2. Three angles of a quadrilateral are equal and the fourth angle is 120∘. Find the measure of the angles.

Answer: The sum of angles in a quadrilateral is 360∘. Let the three equal angles be x. 3x+120∘=360∘ 3x=240∘ x=80∘ (The three angles are each 80∘).

3. Draw a right bisector for a line segment measuring 6.5 cm.

Answer: (Construction task). Draw line AB=6.5 cm. Use a compass set to more than half the length, draw arcs from A and B above and below the line. Connect the intersection points of the arcs.

4. Draw, label and explain the following: (a) Tetrahedron, (b) Triangular Prism

Answer: * Tetrahedron: A triangular pyramid. It has 4 faces, all of which are triangles.

Triangular Prism: A prism with two triangular bases and three rectangular sides.

5. A quadrilateral has three angles measuring 80∘. Find the fourth angle.

Answer: 360∘−(80∘+80∘+80∘)=360∘−240∘=120∘.

6. Write the number of faces, edges and vertices for the following: | Shape | Faces | Edges | Vertices | | :--- | :--- | :--- | :--- | | Cuboid | 6 | 12 | 8 | | Prism (Triangular) | 5 | 9 | 6 | | Square Pyramid | 5 | 8 | 5 | | Cylinder | 3 | 2 | 0 |

7. The angles are in the ratio of 3:5:7:9. Find the measure of the angles.

Answer: Sum of ratios =3+5+7+9=24. Total degrees =360∘.

Angle 1: (3/24)×360=45∘
Angle 2: (5/24)×360=75∘
Angle 3: (7/24)×360=105∘
Angle 4: (9/24)×360=135∘

8. Draw a line AB. Take a point P outside it. Draw a line passing through P perpendicular to AB.

Answer: (Construction task). Place compass on P, draw an arc cutting AB at two points. From those two points, draw arcs below AB that intersect. Connect P to that intersection.

9. Draw and Differentiate between Convex and Concave quadrilateral.

Answer: In a Convex quadrilateral, all interior angles are less than 180∘. In a Concave quadrilateral, one interior angle is greater than 180∘ (a reflex angle).

10. When can a square be considered as a special rhombus?

Answer: A square is a special rhombus when it has four right angles (90∘). (Since a rhombus already has four equal sides, adding right angles makes it a square).

Part 2: Multiple Choice & Shapes

11. What is the correct name for the shape? (Image shows adjacent sides equal)

Answer: d. Kite

12. Which one of these quadrilaterals has one pair of opposite, parallel sides?

Answer: b. Trapezium

13. How does a rhombus differ from a parallelogram?

Answer: a. All sides are equal in length

14. What makes an isosceles trapezium different to an 'ordinary' trapezium?

Answer: b. Non-parallel lines of equal length (and a. A line of symmetry is also correct).

15. A 3-D shape has been opened up to a flat 2-D shape (a net) and is shown below? What was the 3-D shape?

Answer: b. Cylinder (The net shows a rectangle and two circles).

16. Which two mathematical shapes could you combine to make this building?

Answer: c. Cone and cylinder

17. Draw a line segment AB measuring 6.4 cm. Take a point P on AB such that AP is 3.2 cm. Draw a line passing through P perpendicular to AB.

Answer: Note that 3.2 is exactly half of 6.4, so P is the midpoint. This task is essentially drawing a perpendicular bisector at the midpoint of the line.

18. Draw an isosceles trapezium. Draw and write the measurements of the diagonals.

Answer: In an isosceles trapezium, the diagonals are always equal in length. If you draw one where the top is 4 cm and the bottom is 8 cm, your diagonals will measure the same length as each other.

Would you like me to explain the steps for any of the compass constructions (like Question 1 or 8) in more detail?

Certainly! Constructing angles and lines using only a ruler and compass is all about using circles and arcs to find exact points. Here is a step-by-step guide for those specific constructions.

Question 1: Constructing Special Angles

All of these start with a baseline and an initial arc.

a. Constructing

1. Draw a straight line and mark a point O in the middle.

2. Place the compass on O and draw a semi-circle that cuts the line at points A and B.

3. Open the compass slightly wider. Place it on A and draw an arc above the line.

4. Keeping the same width, place the compass on B and draw another arc that crosses the first one at point C.

5. Draw a line from O through C. This is your angle.

c. Constructing (Bisecting )

1. Follow the steps above to make a angle.

2. Place the compass point where the semi-circle hits the baseline and draw an arc in the "corner."

3. Place the compass point where the semi-circle hits the vertical line and draw another arc that crosses the previous one.

4. Draw a line from the origin through this cross.

d. Constructing

1. 150∘ is exactly halfway between and .

2. Draw a baseline and a semi-circle from point O.

3. Without changing the compass width, mark (first arc on the semi-circle) and (second arc).

4. Bisect the space between the mark and the end of the line.

5. The resulting line from O creates a angle from the starting point of the baseline.

Question 8: Perpendicular from an External Point

This method allows you to drop a perfectly "square" line from a point P that isn't sitting on the line itself.

The Steps:

1. The "Smile" Arc: Place your compass sharp point on point P. Open it wide enough so it can reach past the line AB. Draw a wide arc that cuts the line AB at two different spots. Let’s call these spots X and Y.

2. The Crossing Arcs: * Move the compass point to X and draw an arc below the line.

o Move the compass point to Y (keep the same width!) and draw another arc that crosses the one you just made.

3. The Final Line: Use your ruler to connect point P to the spot where those two lower arcs crossed.

This line will hit AB at exactly .

Would you like me to walk you through the steps for any of the other constructions, like the ∘ angle or the ∘ angle?

In geometry, quadrilaterals are classified as either convex or concave based on how their angles and diagonals behave.¹

Visual Comparison

Getty Images

1. Convex Quadrilateral

A convex quadrilateral is the "standard" shape you usually see (like a square or rectangle).

· The "Angles" Rule: All four interior angles are less than ² $180^{\circ}$ .³

· The "Diagonal" Rule: Both diagonals lie entirely inside the shape.⁴

· The "Look": It looks "blown out" or full; no part of the boundary points toward the center.

2. Concave Quadrilateral

A concave quadrilateral (often called a "dart" or "arrowhead") has a "caved-in" appearance.⁵

· The "Angles" Rule: One interior angle is a reflex angle (greater than ⁶$180^{\circ}$).⁷

· The "Diagonal" Rule: At least one diagonal lies outside the shape.⁸

· The "Look": It has a "dent" or a "cave" where one vertex points inward.⁹

Key Differences at a Glance

Feature	Convex Quadrilateral	Concave Quadrilateral
Interior Angles	All four angles are $< 180^{\circ}$ .	One angle is $> 180^{\circ}$ (reflex).
Diagonals	Both diagonals are inside.	One diagonal is outside.
Shape	Points outward; no "dents."	Points inward; contains a "cave."
Examples	Square, Rectangle, Rhombus.	Arrowhead, Dart.

Would you like me to draw a specific example of each, or perhaps show you where the diagonals would sit on a concave shape?

Convex and Concave Quadrilaterals

This video provides a helpful visual breakdown of the differences between convex and concave quadrilaterals using clear examples.

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