Frames:

Extension and investigation

Here are some investigations and problems for yourself, that that you can also use with your learners to reinforce the concept of similarity and its applications.

Golden triangle and pentagon
Investigate this beautiful figure … Click on the picture to open the Pentagram applet.

ABCDE is a regular pentagon, i.e. all its sides and all its angles are equal. The diagonals are drawn, resulting in a star called a pentagram and an embedded similar smaller pentagon PQRST. This process can be continued infinitely …
  1. Calculate the relevant angles in the figure. Name a few congruent and a few similar triangles.
  2. Click on the Golden triangle 1 button. Prove that DDAB is a 36°-72°-72° triangle. Now prove that:

    Tip: Put DT = AB = a and TB = b and prove the golden ratio structure, i.e..
    We call DAB a golden triangle. Like the golden rectangle, we can construct a sequence of golden triangles – show that triangles BAT and TBS are golden triangles. How many golden triangles can you find in our pentagon?
  3. Click on the Golden triangle 2 button. Prove that DDEC is a 108°-36°-36° triangle. Now prove that:

    DDEC is also a golden rectangle, different from DDAB. How many such golden triangles can you find in our pentagon?
  4. Click on the Other ratios button. The many parallel lines lead to similar triangles and interesting relationships. For convenience, let d1 and s1 denote the diagonal and side of the first pentagon ABCDE, and d2 and s2 the diagonal and side of the second pentagon PQRST, etc. Prove that:

Properties of triangles
Click on the picture to open an applet of similar triangles ABC and A¢B¢C¢. You can change the shape of the triangles by dragging A, B and C and change the scale factor by dragging A¢.

Investigate the relationship between the following properties of the two triangles and the ratio of the sides of the triangles (the scale factor): perimeter, area, altitude, median. Prove your conjectures!


Tangents
Click on the picture to open an applet showing three circles touching internally.

Investigate the ratio of the length of the sides of the triangles formed by two chords. How is this related to the radii of the circles?

Can you explain or prove your findings?

Chords, secants and tangents
Click on the picture to open the Chords applet.

  1. Chords AB and CD intersect at P inside the circle.
    Prove that PA.PD = PB.PC
  2. Specialise by checking the nature of the result when:
    1. P is the centre of the circle.
    2. PA = PB and PC = PD (when AB || CD).
  3. Generalise by proving that the relationship in 1 applies also when AD and BC intersect outside the circle (interchange B and D in the applet).
  4. Now specialise by taking the following limiting cases. How does the result in 1 change?
    1. Secant PDA becomes a tangent PA to the circle (move A and D to fall on top of each other).
    2. Secant PBC also becomes a tangent PB to the circle.
    5. Show that you can deduce that PA = PB.



A star
In this applet, three lines are drawn parallel to the sides of DABC.

Drag the red points ...

Are all the smaller triangles formed similar?

Prove it!