Golden Thoughts

Rectangle PQRS has X and Y on the edges. Triangles PQY, YRX and XSP have equal areas. Prove X and Y divide the sides of PQRS in the golden ratio.

From All Corners

Straight lines are drawn from each corner of a square to the mid points of the opposite sides. Express the area of the octagon that is formed at the centre as a fraction of the area of the square.

Star Gazing

Find the ratio of the outer shaded area to the inner area for a six pointed star and an eight pointed star.

Triangle in a Triangle

Stage: 4 Challenge Level:

This problem comes in two parts.

We will be exploring area properties of triangles within triangles, and along the way using some geometric reasoning based on area and proportion.

Part 1.

Take any triangle. Construct a point on each side so that the side length is divided in the ratio $1:2$

Now form a triangle by joining up these points.

Can you work out the fraction of the original triangle that is covered by this new inner triangle?

Part 2.

Now for a different kind of inner triangle.

New start - back to the original triangle with points located on each side as before.

Lines are now drawn across the original triangle from each vertex to those points on the side opposite.
These lines enclose a triangular region (yellow) whose area is always $\frac{1}{7}$ the area of the original triangle.

Can you see why that should be so?

There are plenty of other relationships to go after in this arrangement:

The interior lines divide each other in the ratio $3:3:1$.

Try to see why.

(I've explained that in the Hint section using Vectors. If you'd like to see that - it's a very powerful technique and not too hard to follow)

What else? The blue triangles are equal to each other in area, and the orange quadrilaterals are also equal to each other in area.

Which do you think might be the easiest of all these relationships to make a start with?

Final footnote

Do you know how to replicate (copy) any angle using only a pair of compasses, instead of a protractor?

You'll need to know that if you want to divide a line in a given ratio, for example $1:2$.

Look in the Hint section if you want to check.