Foundations path · F07

Before this lesson: No earlier lesson is required. Start here and take your time.

Your goal: Read only the given geometric facts and use a consistent ratio convention.

Start with the idea and one example. Take a break before the written task if you need it. The questions are original teaching exercises, not official past-paper questions.

Geometry Basics: Lines, Angles and Ratios: the key idea

A diagram records relationships, not measurements you may assume. Name points before using them, distinguish a segment from its full line, and mark only given or proved equalities. An angle names its vertex in the middle: ∠ABC has vertex B. Ratios compare lengths, while areas scale as the square of a length scale. A triangle has area bh/2, where h is perpendicular to the chosen base; the foot of that perpendicular may lie outside an obtuse triangle.

A worked example

Two triangles have the same altitude and bases 4 and 7. Find their area ratio.

Their areas are 4h/2 and 7h/2. Dividing cancels the common nonzero altitude and the factor 1/2, giving 4:7. The result comes from the shared height, not from how the drawing looks.

Your turn: change one thing

An enlargement doubles all lengths. What happens to a triangle’s area?

Try this on paper before opening the explanation.

Compare your reasoning

Its base and perpendicular height both double, so the new area is (2b)(2h)/2 = 4(bh/2). The area becomes four times the original.

Pause and check

A trap to avoid: State the domain, keep exact values and explain why each step is allowed. A correct answer still needs a reason.

Practise and adjust the level

Foundation checks the language; Core applies the method; Stretch asks you to choose or justify an idea. These are levels within this lesson. A session selects six of the nine questions; unused questions allow the level to change. Advanced theory still needs written practice.

Interactive practice loads here. You can also use the complete question set below.

Write a complete argument

Prove that a median divides a triangle into two equal areas.

Planning hint

List the assumptions and the exact conclusion. Identify the definition or theorem in this lesson that connects them. Explain why its conditions hold before using it.

Read the full solution after your attempt

Let D be the midpoint of BC in nondegenerate triangle ABC. Triangles ABD and ACD have equal bases BD=DC on the same line and the same perpendicular height from A. The formula base×height/2 therefore gives equal areas. No equality of AB and AC is needed.

My proof notebook

Write on paper, or keep a draft here. Compare your reasoning with the solution only after a real attempt. The checklist is your own review, not an automatic mark.

All nine practice questions

Prefer paper or have JavaScript switched off? The complete question set, hints and solutions are here. Interactive practice uses these same questions in an order chosen from your answers.

1. Which point is the vertex of ∠ABC?

Foundation

  1. C
  2. Any of the three
  3. B
  4. A
Hint

The middle letter names the vertex.

Answer and reasoning

B. The rays BA and BC meet at B.

2. A triangle has base 8 and perpendicular height 5. Find its area.

Foundation

  1. 10
  2. 20
  3. 40
  4. 13
Hint

Use half the product.

Answer and reasoning

20. Area = 8×5/2 = 20.

3. Two triangles share a height and have bases in ratio 2:5. What is their area ratio?

Core

  1. 4:25
  2. 5:2
  3. 1:1
  4. 2:5
Hint

Write both areas with the same h.

Answer and reasoning

2:5. The common height and factor 1/2 cancel, leaving the base ratio.

4. A similar figure has length scale 3. What is its area scale?

Core

  1. 27
  2. 9
  3. 3
  4. 6
Hint

Both independent length directions scale.

Answer and reasoning

9. Base and height each multiply by 3, so area multiplies by 3²=9.

5. A sketch looks isosceles but no equal sides are given or proved. What may you conclude?

Stretch

  1. The base angles are equal
  2. The median is an altitude
  3. All angles are 60°
  4. No side equality from appearance alone
Hint

Separate given information from appearance.

Answer and reasoning

No side equality from appearance alone. The diagram may be distorted or only approximate. A proof needs a stated or derived equality.

6. Why do triangles with equal bases not necessarily have equal areas?

Stretch

  1. Their perpendicular heights can differ
  2. Areas never use bases
  3. The triangle inequality forbids it
  4. Their angles must all differ
Hint

Check both factors in bh/2.

Answer and reasoning

Their perpendicular heights can differ. Equal bases guarantee equal areas only when the corresponding heights are also equal.

7. Which symbol names the angle with vertex B?

Foundation

  1. ∠ABC
  2. ∠BAC
  3. ∠ACB
  4. AB
Hint

The middle letter is the vertex.

Answer and reasoning

∠ABC. In ∠ABC the rays are BA and BC.

8. A triangle has base 8 and perpendicular height 5. Its area?

Core

  1. 10
  2. 20
  3. 40
  4. 13
Hint

Use half the base-height product.

Answer and reasoning

20. 8·5/2=20 square units.

9. Can a picture alone justify that two lines are parallel?

Stretch

  1. Only when drawn in green
  2. Only on a large screen
  3. No, use given information or a proof
  4. Yes if they look parallel
Hint

A sketch is not a premise.

Answer and reasoning

No, use given information or a proof. Angle equalities or other stated conditions must establish parallelism.

Choose your next step

Continue to Sequences and sums. If this felt difficult, return to a prerequisite above. Every lesson stays open.

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