Maths circle: Pythagorean theorem – We Solve Problems

Maths circle: Pythagorean theorem

Today we will focus on applications of Pythagorean theorem, in geometry and number theory. This famous and ancient theorem states that in a right triangle, the area of a square build on a hypothenuse (the longest side) is a sum of areas of squares build on the other two sides.
$$a^2 + b^2 = c^2$$
$\\$ There are over a 100 proofs of Pythagorean theorem, a quite simple one is visible below:

Four right triangles in this picture are identical: $\triangle A A’ D, \triangle EFA’, \triangle GFH$ and $\triangle CHD$. By moving the triangles around you can see that the large red square has the same area as the sum of areas of the other two squares, violet and green. $\\$
Today’s session is not only about geometry, we will also learn something about the equation $a^2 + b^2 = c^2$, where all the numbers $a,b,c$ are integers.

Problems:

1.

Pythagorean theorem and its converse , Triangles , Types of triangles

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?

2.

Pythagorean theorem and its converse , Triangles , Types of triangles

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

3.

Pythagorean theorem and its converse , Triangles , Types of triangles

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?

4.

Pythagorean theorem and its converse , Triangles , Types of triangles

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 $AD^2 = BD \times AC$ in this particular case. Do you think it is true in general?

5.

Pythagorean theorem and its converse , Triangles , Types of triangles

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.

6.

Pythagorean theorem and its converse , Triangles , Types of triangles

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.

7.

Pythagorean theorem and its converse , Triangles , Types of triangles

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

8.

Circles , Pythagorean theorem and its converse , Triangles (other)

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

9.

Pythagorean theorem and its converse , Triangles , Types of triangles

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 $a^2 +b^2 = c^2$.

10.

Pythagorean theorem and its converse , Triangles , Types of triangles

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?

11.

Pythagorean theorem and its converse , Triangles , Types of triangles

The lengths of three sides of a right triangle are all integer numbers. $\\$
a) Show that one of them is divisible by $3$. $\\$ b*) Show that one of them is divisible by $5$.

12.

Pythagorean theorem and its converse , Triangles , Types of triangles

A segment $AB$ is a base of an isosceles triangle $ABC$. A line perpendicular to the segment $AC$ was drawn through point $A$ – this line crosses an extension of the segment $BC$ at point $D$. There is also a point $E$ somewhere, such that angles $\angle ECB$ and $\angle EBA$ are both right. Point $F$ is on the extension of the segment $AB$, such that $B$ is between $A$ and $F$. We also know that $BF = AD$. Show that $ED =EF$.

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