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Matter of Scale

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Two complete solutions were received from Charles Blackham (Age 13) for Shrewsbury School and Andrei Lazanu (Age 12) from School No 205, Bucharest, Romania. I have used the solution from Andrei and the diagram from Charles below. For those of you who find it difficult to see how the similar triangles match - it is a good idea to redraw the smaller triangle so that it is in the same orientation as its enlargement. This helps you to match the corresponding sides and apply the enlargement scale factor. Well done Andrei and Charles.

The triangle T has sides of length a, b and c and angles x, yand 90 o . Two enlargements are made of triangle T. The triangle T a , is an enlargement by scale factor a and the triangle T b is an enlargement by scale factor b. The triangles T a and T b are fitted together as shown in the diagram. Prove that the resulting triangle is an enlargement of triangle T by the scale factor c and use this fact to prove Pythagoras Theorem.

I first observe that triangles ABD, ADC and ABC are all similar with the original triangle of sides a, b and c. Triangle ABD represents the enlargement by the scale factor a, and triangle ADC the enlargement by the scale factor b.

Solution.

Let x be the acute angle opposed to the side of length a, y the angle opposed to the side of length b. Their sum is 90 o , because the triangle of sides a, b and c is a right angled one. Triangle ABC is also a right angled one, because the measure of angle BAD is x, and the measure of angle DAC is y, and their sum is 90 o . On the other side, the measure of angle ABD is y, and of angle ACD is x. This means triangle ABC is similar with the original triangle.

Now, using the similarity ratio of the original triangle with triangle ABD; I obtained the lengths of the sides:

AB = ac
AD = ab
BD = a 2

Using the similarity ratio of the original triangle with triangle ADC; I obtained the lengths of the sides:

AC = bc
DC = b 2
AD = ab

In the triangle ABC, I already know that AB = ac, and AC = bc. Using the similarity of triangles ABC and the original one, I obtained that BC has length c 2 .

But BC is composed from the segments BD and DC, so the following relation can be written:

BC = BD + DC
c 2 = a 2 + b 2

that is the proof of the Pythagorean Theorem.