Problems

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Matt built a simple wooden hut to protect himself from the rain. From the side the hut looks like a right triangle with the right angle at the top. The longer part of the roof has 20 ft and the shorter one has 15 ft. What is the height of the hut in feet?

The three sides of a right triangle have all integer lengths. Show that at least one of them has an even length.

A bamboo tree, originally 32 metres high, broke in two parts. The end of the other one has fallen 16 metres far from the trunk. How high is the remaining vertical part of the bamboo tree?

Matt has build an additional support for his hut (AD), whose length is equal to the height of the hut calculated in the Example 1. What are the distances from the base of the support to both ends of the hut? Looking at the picture, what are the distances BD and DC, if AB=20 and AC=15? Show that AD2=BD×AC in this particular case. Do you think it is true in general?

Can you build a right triangle using 25 identical matches, without breaking them? You have to use them all. If yes, show how, if not, show why.

On the sides of a right triangle three equilateral triangles were built. Show that the areas of two of the two smaller ones sum up to the area of the larger one.

One triangle has sidelengths 25,25 and 48, the other triangle has sidelengths 25,25 and 14. Which of them has a larger area?

A triple of natural numbers a,b,c such that a2+b2=c2 is called a Pythagorean triple. There are some small Pythagorean triples that are well-known, like 3,4,5 and 5,12,13. Let us have a look at the latter one. We can notice an interesting thing: not only 52+122=132, but also 52=25=12+13 and 1312=1. Use that as an inspiration to find an idea of how to generate some more Pythagorean triples. Check if they are correct by plugging them into the equation a2+b2=c2.

Two semicircles and one circle were drawn on the sides of a right triangle. The circle whose centre is in the midpoint of the hypothenuse actually goes through the right angle corner – this is a general fact, but you don’t need to prove it here. If the two shorter sides of the triangle are 3 and 4, what is the total area of the red region?