Largest square in the corner of a right-angled triangle
Largest square in the corner of a right-angled triangle
IP 2
8 marks
Triangle \(ABC\) has \(\angle C=90^\circ\), \(CA=b\) cm and \(CB=a\) cm. A square \(CPQR\) of side \(s\) cm is drawn with \(P\) on \(CA\), \(R\) on \(CB\) and \(Q\) on \(AB\), as shown in the diagram.
- By considering triangles \(APQ\) and \(ACB\), show that \(s=\dfrac{ab}{a+b}\).[3]
- Find \(s\) when \(a=3\) and \(b=6\).[1]
- Prove that, for every such triangle, the square covers at most half of the area of the triangle, and state when it covers exactly half.[4]
Solution:
Solution locked
680+ questions here come with free worked solutions. A free account saves your questions and progress; this one needs tuition enrolment.
Create a free account to practise similar questions.
Answer:(a) Shown (b) \(2\) cm (c) Ratio \(\dfrac{2ab}{(a+b)^2}\le\dfrac{1}{2}\) since \((a-b)^2\ge0\); exactly half when \(a=b\)
Need help? Join our IP Math tuition programme.
Learn more