Linear algebra (Osnabrück 2024-2025)/Part II/Exercise sheet 36

Exercises

Exercise

Recall concepts and rules for angles (adjacent angle, vertical angle, corresponding angle, alternate angles). Prove these properties in an elementary-geometric way, and with vector geometry.

Exercise

Recall the concepts acute triangle, obtuse triangle, equilateral triangle, and isosceles triangle.

Exercise

Show by elementary-geometric means that the sum of angles in a triangle equals ${}180$ degree.

Exercise

Show that in a nondegenerate triangle, there is at most one right angle.

Exercise

Let two nondegenerate triangles ${}\Delta _{1}=(A_{1},B_{1},C_{1})$ and ${}\Delta _{2}=(A_{2},B_{2},C_{2})$ in affine planes ${}E_{1}$ and ${}E_{2}$ be given. Show that there exists a bijective affine mapping

$\varphi \colon E_{1}\longrightarrow E_{2}$

that transforms one triangle into the other.

Exercise

Show that under a translation in the Euclidean plane, the side lengths and the angle of a triangle do not change.

Exercise

Let ${}D_{1}$ and ${}D_{2}$ be triangles with the property that two side lengths and the angles between them are the same. Show that the two triangles are congruent.

Exercise

Let ${}D_{1}$ and ${}D_{2}$ be triangles with the property that a side length and the two adjacent angles at this side coincide. Show that the two triangles are congruent.

Exercise

Show that an isosceles triangle is congruent to a triangle if and only if it is properly congruent to the triangle. Show moreover that a (not isosceles) triangle might be congruent to a triangle but not properly congruent to it.

Exercise

Let ${}P_{1}=(a_{1},b_{1}),\,P_{2}=(a_{2},b_{2})$, and ${}P_{3}=(a_{3},b_{3})$ be three points in ${}\mathbb {R} ^{2}$. Express the area of the corresponding triangle by a formula involving ${}a_{1},b_{1},a_{2},b_{2},a_{3},b_{3}$.

Exercise

Let three points ${}P_{1},P_{2},P_{3}\in \mathbb {Q} ^{2}\subset \mathbb {R} ^{2}$ be given. Show that the area of the triangle given by these three points is a rational number.

Exercise

Show that in the Euclidean plane, two triangles are similar to each other if and only if their angles coincide.

Exercise

What elementary-geometric proofs of the Pythagorean Theorem do you know?

Exercise

Let ${}A,B,C$ be a right triangle, with the right angle at the point ${}C$. Show that the foot of the altitude through ${}C$ lies on the line segment ${}{\overline {A,B}}$.

Exercise

Determine for the triangle in ${}\mathbb {R} ^{2}$, given by the vertices ${}(0,0),(3,0),(0,5)$, its side lengths, parameter representations for the altitude lines, the lengths of the altitudes, and the feet of the altitudes.

Exercise *

a) Give an example of rational numbers ${}a,b,c\in {]0,1[}$ such that

${}a^{2}+b^{2}=c^{2}\,.$

b) Give an example of rational numbers ${}a,b,c\in {]0,1[}$ such that

${}a^{2}+b^{2}\neq c^{2}\,.$

c) Give an example of irrational numbers ${}a,b\in {]0,1[}$ and a rational number ${}c\in {]0,1[}$ such that

${}a^{2}+b^{2}=c^{2}\,.$

A Pythagorean triple is an integral solution ${}(x,y,z)\in \mathbb {Z} ^{3}$ of the Diophantine equation

${}x^{2}+y^{2}=z^{2}\,.$

It is called primitive if ${}x,y,z$ do not have a common factor ${}\neq 1$.

Exercise

Let ${}x$ and ${}y$ be odd. Show that ${}x^{2}+y^{2}$ is not a square.

Exercise

Let ${}(x,y,z)$ be a Pythagorean triple. Show that ${}x$ or ${}y$ is a multiple of ${}3$.

Exercise

Sketch a triangle ${}D$ such that one altitude divides the triangle ${}D$ into two different right triangles ${}D_{1}$ and ${}D_{2}$ in such a way that the side lengths of ${}D_{1}$ and of ${}D_{2}$ form Pythagorean triples. Give also the side lengths.

Exercise

Let ${}A,B,C$ be a right triangle, with the right angle located at ${}C$. Let ${}h$ be the altitude through ${}C$, and let ${}D$ denote the foot of this altitude through the linen given by ${}A$ and ${}B$. Show that

${}d(C,D)^{2}=d(A,D)\cdot d(B,D)\,$

holds.

Exercise

Prove the reverse statement of Thales's theorem: Let ${}A,B,C$ be a right triangle, with the right angle at ${}C$. Let ${}M$ be the midpoint of the line segment ${}{\overline {A,B}}$. Then

${}d(C,M)=d(A,M)=d(B,M)\,,$

that is, ${}C$ belongs to the circle with center ${}M$ through ${}A$ (and ${}B$).

Exercise

Let a triangle ${}(A,B,C)$ be given, with side lengths ${}a,b,c$. Let ${}\gamma$ denote the angle at ${}C$. Show that

${}c^{2}=a^{2}+b^{2}-2ab\cos \gamma \,$

holds.

Exercise

Prove the reverse statement of the Pythagorean theorem: If in a triangle the relation

${}c^{2}=a^{2}+b^{2}\,$

between the side lengths ${}a,b,c$ holds, then the triangle is a right triangle.

Exercise

Let ${}M$ be a metric space consisting of three points. Show that ${}M$ can be realized as a metric subspace of a Euclidean plane.

Exercise

Let ${}A,B,C$ be a triangle in the Euclidean plane, and let ${}R$ denote the boundary of the triangle, that is, the union of the three edges.

a) Define a metric on ${}R$ such that the distance of two points on the same edge is just the induced distance, and such that

$d(x,y)$

is the minimal distance along a path on ${}R$ connecting ${}x$ and ${}y$.

b) Is this the induced metric?

c) Is it possible that the shortest connection between two points uses all three edges?

In the following exercises, we develop the concept of a convex hull.

A subset ${}T\subseteq \mathbb {R} ^{n}$ is called convex if for any two points ${}P,Q\in T$, the line segment between the two points, that is, every point of the form

$rP+(1-r)Q{\text{ with }}r\in [0,1],$

also belongs to ${}T$.

For a subset ${}U\subseteq \mathbb {R} ^{n}$, the smallest convex subset

${}T$ that contains ${}U$ is called the convex hull of ${}U$.

Exercise

Are all quadrilaterals convex?

The existence of the convex hull rests on the following observation.

Exercise

Show that the intersection of convex subsets in ${}\mathbb {R} ^{n}$ is again convex.

Exercise

Let ${}P_{1},\ldots ,P_{m}$ be points in ${}\mathbb {R} ^{n}$. Show that the convex hull of these points equals the set of all non-negative barycentric combinations, that is,

${\left\{\sum _{i=1}^{m}a_{i}P_{i}\mid \sum _{i=1}^{m}a_{i}=1,\,a_{i}\geq 0{\text{ for all }}i\right\}}.$

Exercise

Divide geometrically the given line segment into five parts of the same length.

Exercise

Prove that in the situation of Theorem 36.16 , there are similar triangles.

Hand-in-exercises

Exercise (5 marks)

Give, for the triangles

$(2,1),(2,-1),(5,-1)\,\,{\text{ and }}\,\,(-1,1),(1,1),(1,4),$

an explicit sequence of translations, rotations, and reflections at a line, that transforms one triangle into the other.

Exercise (4 marks)

Determine for the triangle in ${}\mathbb {R} ^{2}$, given by the vertices ${}(2,-3),(4,1),(5,6)$, its side lengths, parameter representations for the altitude lines, the lengths of the altitudes, and the feet of the altitudes.

Exercise (8 (2+1+1+4) marks)

In ${}\mathbb {R} ^{3}$, let the triangle with the vertices ${}(4,2,-5),(4,3,7),(-5,0,-6)$ be given.

a) Determine an equation and a parameter representation of the affine plane, in which the triangle lies.

b) Determine the side lengths of the triangle.

c) Determine the angles of the triangle.

d) Determine a parameter representation of the altitude line through the point ${}(4,2,-5)$, the length of this altitude, and the corresponding feet of the altitude.

Exercise (4 marks)

Let ${}A,B,C$ be a triangle in a Euclidean plane. Show that the distance between the vertex ${}C$ and the edge ${}{\overline {AB}}$ is obtained in the point ${}A$, or in the point ${}B$, or in the foot of the altitude of the altitude through ${}C$.

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