Problems

Author: A.V. Shapovalov

We call a triangle rational if all of its angles are measured by a rational number of degrees. We call a point inside the triangle rational if, when we join it by segments with vertices, we get three rational triangles. Prove that within any acute-angled rational triangle there are at least three distinct rational points.

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 $A{D}^2=BD\times AC$ 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?

Find the area of the green annulus, if the marked pink segment has length $1$.

A triple of natural numbers $a,b,c$ such that $a^2+b^2=c^2$ 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 $5^2+{12}^2={13}^2$, but also $5^2=25=12+13$ and $13-12=1$. 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 $a^2+b^2=c^2$.