Wedge on wedge
Two right-angled triangles are connected together as part of a
structure. An object is dropped from the top of the green triangle
where does it pass the base of the blue triangle?
Age
14 to 16
Challenge level
Problem
Two right-angled triangles are connected together as part of a structure.
Image
The green triangle has side lengths of 65, 52, and 39 cm and the blue triangle has side lengths of 52, 48, and 20 cm.
An object is dropped from the top of the green triangle, where does it pass the base of the blue triangle?
In what proportion is the blue base split and how far has the object fallen?
This problem does not require a calculator or any special formula.
Getting Started
Draw in the path of the falling object and look at the triangles that are created within your diagram - be prepared to add an extra horizontal line maybe.
Student Solutions
Image
We contruct a new triangle as shown.
Angle $x$ is equal to $90^\circ - (90^\circ - y) = y$, so the
highlighted triangles are similar. Therefore $$p = 48 \times
\frac{39}{52} = 36$$ $$q =20\times \frac{39}{52} = 15$$
The base is split $48 - q : q = 33 : 15 = 11 : 5$ and the ball falls a distance of $p + 20 = 36 + 20 = 56$.
Teachers' Resources
This problem, based on a structure that includes a compound angle, could be a standard post-16 trigonometry question, but offered here, with side lengths selected to make ratio calculations easy, we hope to encourage students to explore the construction on which the compound angle formulae rest.
The numbers were chosen to make calculation as easy as possible. It seemed that once students reached for a calculator to help them multiply or divide, they would think that the SIN, COS or TAN buttons were the answer here and this is not a terribly useful directon to take.
But there may be value in asking students how they might solve the problem with numbers that they make up for themselves (or for each other). Can they describe ageneral strategy? It is also useful at this point to ask whether all the side length data is necessary.
A short distance beyond Stage 4 students will have a range of formulae to apply to problems like this, our aim at this stage is to help them spend some time exploring the constructions that make those formulae possible or valid.