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# Pythagorean theorem and orthogonality
One of the first geometric fact that you may have learned is the Pythagorean theorem, which states:
> **Theorem.** Pythagorean theorem.
> If a right triangle has side lengths $a,b,c$ with $c$ as the side length of the hypotenuse, then we have $a^{2}+b^{2}=c^{2}$.
In fact the **converse** is also true:
>**Theorem.** Converse to Pythagorean theorem.
>If a triangle has side lengths $a,b,c$ such that $a^{2}+b^{2}=c^{2}$, then the triangle is a right triangle.
These are mathematical statements. Generally what we want to be able to do is (1) know what it is saying, (2) how to apply it, and (3) understand why it is true.
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## 1 Am I a Triangle?
Given the following three lengths, which of them would form a right triangle?
3-4-5 triangle
10-20-30 triangle
12-14-20 triangle
120-209-241 triangle
1.5-11.2-11.3 triangle
Did you use the Pythagorean theorem or its converse to deduce these?
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## 2 Make right angle with a rope.
Some believe ancient Egyptians actually would use a rope to create a right angle at any location $P$, pointing in any direction they desire. How might that work? Can you do it using just rope (and something to cut the rope), and nothing else?
![[1 teaching/summer program 2023/week 1/---files/Pythagorean_theorem_and_orthogonality 2023-05-09 12.40.08.excalidraw.svg]]
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Note, you can make knots and mark equal lengths... Try it with some yarn !
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## 3 What even makes a triangle?
Would you consider a shape with side lengths 10-20-30 a triangle? Why or why not?
Given three side lengths $a,b,c$, do they always form a triangle? Can you come up with a condition or conditions that will guarantee three lengths $a,b,c$ will form a triangle?
![[1 teaching/summer program 2023/week 1/---files/Pythagorean_theorem_and_orthogonality 2023-05-09 11.33.34.excalidraw.svg]]
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The conditions you come up with above is something called triangle inequality, and it is useful in detecting whether we have a valid triangle or not !
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## 4 Distance formula on the coordinate plane.
Distance formula on the coordinate plane is a consequence of the Pythagorean theorem. Do you see why? Consider the following two points $P_{1}=(x_{1},y_{1})$ and $P_{2}=(x_{2},y_{2})$ on the coordinate plane. Draw out a right triangle and use the Pythagorean theorem to write an **expression** for its distance $d$ in terms of $x_1 ,y_1 ,x_2, y_2$.
![[1 teaching/summer program 2023/week 1/---files/Pythagorean_theorem_and_orthogonality 2023-05-09 11.37.23.excalidraw.svg]]
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## 5 Square gardens and right angled paths.
Consider the following square gardens, each with some walk path from one corner to the opposite corner. The paths **only make right angle turns**. The length of each segment of the paths are given below. What are the **dimensions** of each of the square gardens? Can you tell which garden is the largest, and which one is the smallest?
![[1 teaching/summer program 2023/week 1/---files/Pythagorean_theorem_and_orthogonality 2023-05-09 11.40.55.excalidraw.svg]]
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Hint: Sometimes its good to rotate ones perspective, and may pretend there are coordinates somewhere...
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## 6 Writing algebraic relations and functions.
For the following, draw some diagrams to help you.
(A) Consider a square with perimeter $P$ and area $A$. Express $P$ as a function of $A$.
(B) Consider a square with diagonal $d$ and area $A$. Express $A$ as a function of $d$.
(C) Consider a rectangle whose sides have a ratio of $1:3$, and let $d$ be the length of its diagonal, and $A$ its area. Express the $A$ as a function of $d$.
(D) Consider a right triangle whose two perpendicular sides have a ratio of $2:5$. Let $P$ be its perimeter and $A$ be its area. Express $A$ as a function of $P$.
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## 7 Let's prove the Pythagorean theorem!
Let us come up with a proof for the Pythagorean theorem, that is, understanding why it is true. This is a statement for any right triangles.
Draw an arbitrary right triangle on some paper, and make 4 copies of it:
![[1 teaching/summer program 2023/week 1/---files/Pythagorean_theorem_and_orthogonality 2023-05-09 11.52.21.excalidraw.svg]]
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Label the corresponding side lengths a,b,c appropriately.
Try to arrange them in a way so that they form a larger square, with an inner empty square, without overlapping the triangles. Something that may look like this:
![[1 teaching/summer program 2023/week 1/---files/Pythagorean_theorem_and_orthogonality 2023-05-09 11.54.51.excalidraw.svg]]
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Label all the corresponding sides, and express the area of larger square in different ways, in terms of the parameters $a,b,c$, as well as using the areas of the triangles.
Can you deduce Pythagorean theorem, $a^{2}+b^{2}=c^{2}$?
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## Discussion.
Now that we have Pythagorean theorem, that in a right triangle, the sides satisfy $a^2 + b^2 = c^2$, what about the **converse**? Assuming a triangle has sides satisfying $a^2 + b^2 =c^2$, why is this a right triangle? Discuss in your group. Are there any assumptions that you need to make?
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## 8 Distance formula again, now in 3d.
Draw a 3-d coordinate system and place an arbitrary point at $P=(a,b,c)$. Show the distance from P to the origin $O=(0,0,0)$ is indeed $OP=\sqrt{a^{2}+b^{2}+c^{2}}$ using the Pythagorean theorem twice.
Hint: Use the following diagram, where N is a point you get by projecting $P$ orthogonally onto the $xy$-plane, and $M$ is the point you get by projecting $N$ orthogonally to the $x$-axis.
Find the coordinates of $M$ and $N$; the distance $ON$, using right triangles you see; and find the length $OP$.
![[1 teaching/summer program 2023/week 1/---files/Pythagorean_theorem_and_orthogonality 2023-05-09 12.00.15.excalidraw.svg]]
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Use your result above to find a general formula for the distance between two points $P_{1}=(x_{1},y_{1},z_{1})$ and $P_{2}=(x_{1},y_{2},z_{2})$ in 3-d space.
Can you generalize this to higher dimensions?
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## 10 Lines with perpendicular slopes on the coordinate plane.
Recall that for a line $L:y=mx+b$, whose slope is $m$, a line perpendicular to $L$ would have a slope of $\displaystyle \frac{-1}{m}$ (negative reciprocal). Let us see why this is the case.
Consider the line $L_{1}:y=mx$, and line $\displaystyle L_{2}:y=\frac{-1}{m}x$, for some $m>0$, say. Draw a diagram.
![[1 teaching/summer program 2023/week 1/---files/Pythagorean_theorem_and_orthogonality 2023-05-09 12.18.29.excalidraw.svg]]
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Label the point $A=(1,m)$, which lies on line $L_{1}$.
Label the point $B=(1,\frac{-1}{m})$ which lies on the line $L_{2}$
Label the point $C=(0,0)$, which lies on both lines.
Draw these three points on the diagram, and note they form a triangle. But is it a right triangle?
To figure this out, find the lengths of $AB$, $BC$, and $CA$. What can you conclude, and how do you know?
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## 11 Using dot product to deduce orthogonality.
A (geometric) vector is an arrow drawn from a starting point to an ending point. If the starting point is not specified, then we can treat the vector as starting from the origin. So a vector $v=\langle3,2\rangle$ is an arrow starting from the origin to the point $(3,2)$.
To check if two vectors $v=\left\langle a,b\right\rangle$ and $w=\left\langle c,d\right\rangle$ make perpendicular angle, an easy way to do so is to compute their dot product $v\cdot w=ac+bd$. If this dot product is 0, then they are perpendicular (orthogonal), otherwise they are not.
**Let us see why this is true !**
Given vectors $v=\left\langle a,b\right\rangle$ and $w=\left\langle c,d\right\rangle$ , the points $(0,0),(a,b),(c,d)$ form a triangle.
**Draw a diagram for this !!**
Find the **lengths of each side** of this triangle, and deduce what happens if $v\cdot w=ac+bd=0$.
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(Just something to ponder for now...what if the triangle is not a right triangle, is there a "Pythagorean theorem" for it?)
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## 12 Height of a triangle.
Consider the following 5-7-10 triangle. Find the altittude perpendicular the side that is length 10.
![[1 teaching/summer program 2023/week 1/---files/Pythagorean_theorem_and_orthogonality 2023-05-09 12.30.48.excalidraw.svg]]
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## 13 Height of the pyramids.
Suppose we have a square base pyramid, where each outer edge has a side length is 1 as shown below. Here the apex of the pyramid is above the middle of the square. What is the height of the pyramid?
![[1 teaching/summer program 2023/week 1/---files/Pythagorean_theorem_and_orthogonality 2023-05-09 12.35.11.excalidraw.svg]]
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What about an equilateral triangle base pyramid, where each outer edge has a side length of 1. Again the apex of the pyramid is above the middle of the base triangle. What is the height of this pyramid? By the way, this shape is called a **regular tetrahedron**.
![[1 teaching/summer program 2023/week 1/---files/Pythagorean_theorem_and_orthogonality 2023-05-09 12.37.16.excalidraw.svg]]
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