Year 9 Pythagoras & Applied Geometry Worksheets

Worked examples

Year 9 pythagoras (applied): worked examples

Three examples, from a standard question to a worded problem. Have a go at each one before you open the solution.

1. Example 1 · Diagonal of a rectangular prism

   A rectangular box has length $12$ cm, width $5$ cm and height $8$ cm. Find the length of the longest diagonal of the box, correct to one decimal place.

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   Key idea: The longest diagonal of a rectangular prism goes from one corner to the opposite corner. Use Pythagoras twice: first on the base, then use that result with the height.

   1. Find the diagonal $d$ of the base using Pythagoras:
      $\begin{gathered}d^2 = 12^2 + 5^2 = 144 + 25 = 169 \\ d = 13 \text{ cm}\end{gathered}$
   2. Use $d$ and the height to find the space diagonal $D$:
      $\begin{gathered}D^2 = d^2 + 8^2 = 169 + 64 = 233 \\ D = \sqrt{233} \approx 15.3 \text{ cm}\end{gathered}$

   Answer: The longest diagonal is approximately $15.3$ cm.

2. Example 2 · Similarity and Pythagoras to find a missing length

   Two right-angled triangles are similar. In the smaller triangle, the two shorter sides are $6$ cm and $8$ cm. The longer of those two sides in the larger triangle is $20$ cm. Find the hypotenuse of the larger triangle.

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   Key idea: Use Pythagoras to find the hypotenuse of the smaller triangle, then use the similarity ratio to scale it up.

   1. Find the hypotenuse of the smaller triangle:
      $c^2 = 6^2 + 8^2 = 36 + 64 = 100 \implies c = 10 \text{ cm}$
   2. The side of length $8$ cm in the smaller triangle corresponds to the side of length $20$ cm in the larger triangle. Find the scale factor:
      $k = \frac{20}{8} = 2.5$
   3. Multiply the smaller hypotenuse by the scale factor:
      $\text{Larger hypotenuse} = 10 \times 2.5 = 25 \text{ cm}$

   Answer: The hypotenuse of the larger triangle is $25$ cm.

3. Example 3 · Real-life problem: ladder against a wall

   A $6.5$ m ladder leans against a vertical wall. The base of the ladder is on horizontal ground and sits $2.5$ m from the base of the wall. A safety rule says the angle the ladder makes with the ground must be between $70°$ and $80°$. Use trigonometry to find the angle the ladder makes with the ground, correct to one decimal place, and state whether the ladder is positioned safely.

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   Key idea: The ladder, wall and ground form a right-angled triangle. Use the cosine ratio, since the adjacent side (ground distance) and hypotenuse (ladder) are known, to find the angle at the base.

   1. Identify the sides relative to the angle at the ground: adjacent $= 2.5$ m, hypotenuse $= 6.5$ m.
   2. Write the cosine ratio:
      $\cos(\theta) = \frac{2.5}{6.5}$
   3. Solve for $\theta$:
      $\theta = \cos^{-1}\!\left(\frac{2.5}{6.5}\right) \approx \cos^{-1}(0.3846) \approx 67.4°$
   4. Compare with the safe range of $70°$ to $80°$: $67.4° < 70°$, so the ladder is outside the safe range.

   Answer: The ladder makes an angle of approximately $67.4°$ with the ground, which is not safe.

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