Pythagorean Real-Life Applications

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📚 Part 1: Multiple Choice & Quick Calculations

1. A 5 m ladder leans against a wall and reaches a height of 4 m. How far is the base of the ladder from the wall? (Show your working briefly.)

1 m

3 m

4 m

2 m

2. A rectangular garden has sides 6 m and 8 m that meet at a right angle. What is the length of the diagonal path across the garden?

9 m

10 m

14 m

8.5 m

3. A TV screen has a diagonal of 55 cm and a height of 33 cm. Calculate the width of the screen to the nearest centimetre. Show working.

4. Tick all situations where using the Pythagorean theorem is appropriate:

Finding the diagonal of a rectangular table

Calculating the shortest walking distance across a rectangular park (straight line)

Finding the area of a right triangle

Determining an angle of elevation directly from two side lengths without a right angle

✏️ Part 2: Applied Problems — Show step-by-step working and justify your answers

5. Ladder safety: A 6 m ladder is placed so its top reaches 5 m up the wall. Calculate the horizontal distance from the wall to the base of the ladder (to 2 decimal places). Is the placement safe if the recommended minimum base distance is 1.5 m? Explain your answer.

6. Triangular garden: Two sides of a right-angled triangular garden measure 12 m and 9 m (these are the legs). A gardener wants to place a straight decorative fence along the hypotenuse. Calculate the length of the fence (give exact value and a decimal to 2 d.p.).

7. Land survey (long distance): A surveyor maps a right-angled plot where one boundary is 0.7 miles and another perpendicular boundary is 0.48 miles. Find the length of the third boundary (hypotenuse) in miles (to 3 decimal places). Show working and comment whether it would be sensible to round before calculating.

8. Design a ramp: A school needs a ramp to reach a platform 0.9 m high. The horizontal run available is 4.2 m. (a) Find the length of the ramp using Pythagoras (to 2 d.p.). (b) In the drawing box, sketch a labelled right triangle showing the rise, run and ramp (hypotenuse). Explain in one sentence why Pythagoras is appropriate here.

WALT (We Are Learning To): Use the Pythagorean theorem to calculate missing sides in real-life right-angled situations, interpret results for safety or design, and justify each step clearly.