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Degree Ceremony

What does Pythagoras' Theorem tell you about these angles: 90°, (45+x)° and (45-x)° in a triangle?

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Logosquares

Ten squares form regular rings either with adjacent or opposite vertices touching. Calculate the inner and outer radii of the rings that surround the squares.

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Ball Bearings

If a is the radius of the axle, b the radius of each ball-bearing, and c the radius of the hub, why does the number of ball bearings n determine the ratio c/a? Find a formula for c/a in terms of n.

So Big

Age 16 to 18 Challenge Level:

Why do this problem?

At first sight there might not seem to be enough information given in this question to find a solution. The first step is to interpret what is known from the geometry given the inscribed circle, and then to evaluate that information in terms of how it might be used to express what is known in terms of the three variables $a$, $b$ and $r$ and how that might be used to give an expression for the area of the triangle.

One of the methods of solving this problem is an application of the formula for tan$(A + B)$ combined with the formula for the area of a triangle.

Possible approach

Set this as an exercise in applying the formula for $tan(A + B)$ or better still, if you want the students to revise their trigonometry, you can make it a more challenging problem solving activity and leave it to the students to decide what they have to use.

Key questions

Can we make a list of everything that is known from the information given?

Can we see a way of using the information given to find the area of the triangle?

Can we add lines to make pairs of congruent triangles?

Can we find the area of the whole triangle in terms of these smaller congruent triangles?

What can we say about the angles at the centre of the circle?

If three of the angles at the centre add up to $180$ degrees what can we say about the tangents of these angles?