A circle touches the lines OA, OB and AB where OA and OB are perpendicular. Show that the diameter of the circle is equal to the perimeter of the triangle
A 1 metre cube has one face on the ground and one face against a
wall. A 4 metre ladder leans against the wall and just touches the
cube. How high is the top of the ladder above the ground?
The area of a regular pentagon looks about twice as a big as the pentangle star drawn within it. Is it?
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
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.
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
, 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
= a 2
+ b 2
that is the proof of the Pythagorean Theorem.