Stage 4 Geometry Worksheet for Age 12 | Portal 2 Math & Problem Solving

Instructions

1. You are testing a Portal 2-style training chamber. Solve each geometry task to unlock the next chamber.
2. Show your working, not just your answer. Use a ruler, protractor, and calculator if helpful.
3. When a question asks you to draw, label all important measurements, points, or angles.
4. Complete the reflection prompts honestly. They are about how you think and learn, not just whether an answer is correct.
5. The Core Chambers build from easier to harder questions. Try the Optional Challenge after completing them.

Helpful reminders

• Rectangle area = length × width
• Triangle area = 1/2 × base × height
• Perimeter is the distance around a shape.
• The angles in a triangle add to 180°.
• In a right-angled triangle: a² + b² = c²
• A reflection flips a shape across a mirror line.

Chamber 1: Portal Pad Coordinates

A test chamber uses a coordinate grid. Point A is at (2, 3), point B is at (8, 3), and point C is at (8, 7).

1. What is the horizontal distance from A to B?

   Answer: ____ units

2. What is the vertical distance from B to C?

   Answer: ____ units

3. Plot A, B, and C on squared paper. Add point D at (2, 7).

4. Join the points in order: A → B → C → D → A. What shape have you made?

   Answer: ____

5. Find the perimeter of the shape.

   Working: ____

   Answer: ____ units

6. Find the area of the shape.

   Working: ____

   Answer: ____ square units

Quick reflection: Which information helped you most: the coordinates, the drawing, or the formulas? Explain.

Chamber 2: The Laser Angle Lock

A laser beam travels along a straight corridor. A second beam turns at a corner.

1. Two angles on a straight line are 63° and x°.

   Calculate x.

   Working: ____

   x = ____

2. The angles in a triangular test area are 48°, 67°, and y°.

   Calculate y.

   Working: ____

   y = ____

3. Draw a triangle with angles of 40°, 60°, and 80°. Label each angle.

4. A rotating platform turns through a right angle. How many degrees is this?

   Answer: ____

5. A portal rotates a platform through 270°. Is this a clockwise or anticlockwise turn if the platform moves three quarter-turns clockwise?

   Answer: ____

Hint: A full turn is 360°, a half-turn is 180°, and a quarter-turn is 90°.

Reflection: What mistake might someone make when adding angles in a triangle? How could you help them avoid it?

Chamber 3: Companion Cube Construction

A square test platform has sides of 6 m. A triangular section is added to one side. The triangle has a base of 6 m and a perpendicular height of 4 m.

1. Find the area of the square platform.

   Working: ____

   Answer: ____ m²

2. Find the area of the triangular section.

   Working: ____

   Answer: ____ m²

3. Find the total area of the platform and the triangular section.

   Answer: ____ m²

4. The square platform has a 1 m wide safety strip around its outside edge. The strip is not included in the 6 m by 6 m platform. Explain why the total area of the safety strip cannot be found simply by calculating 1 × 1.

5. A rectangular storage area is 9 m long and 4 m wide. A wall divides it into two equal rectangles. What is the area of each smaller rectangle?

   Answer: ____ m²

Chamber 4: Moving Platforms

A platform is translated 4 units right and 2 units down. Point P starts at (3, 6).

1. Write the new coordinates of P.

   Answer: P′ = ( __ , __ )

2. Point Q starts at (−2, 5). It is translated 3 units left and 4 units up. Write the new coordinates.

   Answer: Q′ = ( __ , __ )

3. Reflect point R at (5, 2) in the x-axis.

   Answer: R′ = ( __ , __ )

4. Reflect point S at (−3, 4) in the y-axis.

   Answer: S′ = ( __ , __ )

5. On squared paper, draw a triangle with vertices (1, 1), (4, 1), and (1, 3). Reflect it in the y-axis. Write the coordinates of all three reflected vertices.

   First vertex: ____

   Second vertex: ____

   Third vertex: ____

Think like a designer: How is a reflection different from a translation?

Chamber 5: The Long Walk

A maintenance robot travels across a rectangular room. It moves 9 m east and then 12 m north in a straightened route around the room.

1. Draw a right-angled triangle to represent the direct distance from the starting point to the finishing point.

2. Use Pythagoras’ theorem to find the direct distance.

   Working: ____

   Answer: ____ m

3. The robot’s battery can travel 20 m. Will it have enough battery for the direct route? Explain.

4. How much shorter is the direct route than the 9 m + 12 m route?

   Answer: ____ m shorter

5. A second route has perpendicular sides of 5 m and 12 m. Find its direct distance.

   Working: ____

   Answer: ____ m

Chamber 6: Data from the Test Run

A group records how many attempts each person needs to complete a geometry chamber. The example row shows how to calculate the range.

Use these results for Players A–E: 3, 7, 4, 6, 5.

1. Complete the table.

2. What is the mode?

   Answer: ____ attempts

3. Find the mean average.

   Working: ____

   Answer: ____ attempts

4. Find the range.

   Working: ____

   Answer: ____ attempts

5. Which measure, mean or mode, better describes the most common result? Explain.

Final Chamber: Design Your Own Test

Design a small Portal 2-style geometry chamber for another learner.

Your chamber must include:

• One shape with a perimeter or area to calculate
• One angle question
• One movement, translation, or reflection
• One real-world reason why the answer matters

Chamber name: ____

Challenge instructions:

Diagram or coordinate plan:

Answer and working:

Real-world connection:

Optional Challenge: Advanced Test Chamber

A rectangular room measures 15 m by 8 m. A right-angled triangular section with base 8 m and height 6 m is removed from one corner.

1. Find the area of the original rectangle.

   Answer: ____ m²

2. Find the area of the removed triangle.

   Answer: ____ m²

3. Find the remaining area.

   Answer: ____ m²

4. The two perpendicular sides of the removed triangle are 8 m and 6 m. Find the length of the diagonal cut.

   Answer: ____ m

5. Explain one way to check whether your answers are reasonable.

Learning Reflection

Complete the sentences.

1. One geometry idea I used confidently was ____.

2. One question that made me think hard was ____.

3. The strategy that helped me most was ____.

4. If I could try one chamber again, I would change _____ because __.

5. One place outside school where I could use geometry is ____.

Answer Key

Chamber 1

1. 6 units
2. 4 units
3. The plotted points form a rectangle.
4. Rectangle
5. Perimeter = 6 + 4 + 6 + 4 = 20 units
6. Area = 6 × 4 = 24 square units

Chamber 2

1. Angles on a straight line total 180°: x = 180° − 63° = 117°
2. Angles in a triangle total 180°: y = 180° − 48° − 67° = 65°
3. The drawing should show a triangle labelled 40°, 60°, and 80°.
4. A right angle is 90°.
5. Three quarter-turns clockwise is a 270° clockwise turn.

Chamber 3

1. Square area = 6 × 6 = 36 m²
2. Triangle area = 1/2 × 6 × 4 = 12 m²
3. Total area = 36 + 12 = 48 m²
4. The safety strip surrounds the entire outside edge, so it is made of several rectangular sections, not just one 1 m by 1 m square.
5. Total area = 9 × 4 = 36 m². Each half = 36 ÷ 2 = 18 m²

Chamber 4

1. P′ = (7, 4)
2. Q′ = (−5, 9)
3. R′ = (5, −2)
4. S′ = (3, 4)
5. The reflected vertices are (−1, 1), (−4, 1), and (−1, 3), in the same order as the original vertices.

A reflection flips a shape across a line. A translation slides a shape without flipping or turning it.

Chamber 5

1. The diagram should be a right-angled triangle with perpendicular sides of 9 m and 12 m and a diagonal representing the direct route.
2. Direct distance² = 9² + 12² = 81 + 144 = 225. Direct distance = 15 m.
3. Yes. The robot needs 15 m, which is less than its 20 m battery limit.
4. The route around the room is 9 + 12 = 21 m. Difference = 21 − 15 = 6 m shorter.
5. Direct distance² = 5² + 12² = 25 + 144 = 169. Direct distance = 13 m.

Chamber 6

2. Mode = 5 attempts
3. Mean = (3 + 7 + 4 + 6 + 5) ÷ 5 = 25 ÷ 5 = 5 attempts
4. Range = 7 − 3 = 4 attempts
5. The mode better describes the most common result because 5 appears more often than any other number.

Optional Challenge

1. Rectangle area = 15 × 8 = 120 m²
2. Triangle area = 1/2 × 8 × 6 = 24 m²
3. Remaining area = 120 − 24 = 96 m²
4. Diagonal² = 8² + 6² = 64 + 36 = 100. Diagonal = 10 m
5. A reasonable check is to notice that the remaining area must be less than 120 m², and the diagonal must be longer than 8 m but shorter than 8 + 6 = 14 m. Both answers meet these checks.

Final Chamber and Learning Reflection

Answers will vary. A successful response should include the required geometry ideas, accurate working, a labelled diagram or coordinate plan, and a sensible real-world connection. Reflection responses should describe the learner’s own thinking and strategies.