Today’s random geometric construction is the perpendicular line from an end point. I like this one a lot because it utilises a circle theorem.

First, draw your arbitrary line, with an arbitrary dot, ideally near the end point of the line like so:

Next, construct a circle with its center as the dot, and radius as dot-to-edge:

Next draw a diameter from point of intersection furthest from your line endpoint:

And finally, join the diameter to your line endpoint:

The angle in a semi-circle is always 90 degrees. Job done! All that’s left is to colour it in, because colouring maths diagrams is underrated.

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I do not know the answers to the two questions I am about to pose – but I am always interested in seeing how a task might be extended, so here goes – and War & Peace will be starting soon:

Question 1 – If we rotate the original line about the end-point which becomes joined to the centre of the circle, then what happens to the area of the blue triangle?

Question 2 – Is there a relationship to be found regarding the ratio of the blue:orange coloured areas?

Regards

Mike

For your second question, if you consider a circle with a diameter drawn and a point on the circle, then connecting our point with the ends of the diameter will give you a right triangle. moving the point on the circle arbitrarily close to one of the diameter end-points will give us triangles with smaller and smaller areas with 0 as a limit (or even allowing 0 if you accept the degenerate triangle). On the other hand, the largest area will be the triangle with the moving point exactly half-way along the circular arc between the diameter endpoints, making an isosceles right triangle.

Some things that are pretty easy to see:

(1) why this is the triangle with largest area

(2) the relationship between the area of that maximal triangle and the area of the circle