Geometry Assignment — Handout 5510

Geometry Assignment — Book 5.1

Geometry Assignment — Handout 5110

Geometry Assignment List — Chapter 5

Geometry

Chapter 5

1. Handout 5310

2. Handout 5210

3. Handout 5110

4. Book 5.1: 1-10, 22-25, 27-36, 40-46

5. Handout 5510

6. Book 5.3 page 260: 11-13, 37-39

7. Book 5.4: 1, 3, 4, 16, 18, 42, 43

8. Handout 5520

9. Book 5.5: 1, 4, 7, 10, 13, 16, 19, 21, 22, 24, 25, 34, 35

10. Book 5.R: 23, 24, 26, 27, 29, 30-42 page 284: 5, 6

Geometry Notes — The Triangle Inequality Theorem

Triangle Inequality Theorem
The triangle inequality theorem states that any side of a triangle is always shorter than the sum of the other two sides.

in other words, the Triangle Inequality Theorem, or as I sometimes call it, the Triangle Test, the sum of the lengths of any two sides of a triangle is greater than the length of the third side.

Imagine the bottom angles’ vertices are hinges. And the sides of length 10 and 13 can pivot on those hinges. Because the sum of 10 and 13 is greater than 15, the sides of 10 and 13 will come together at a cute little point above the base and a wonderful triangle is form. Ahhh.

Imagine the bottom angles’ vertices are hinges in the above example. And the sides of length 5 and 7 can pivot on those hinges. Because the sum of 5 and 7 is less than 15, the sides of 5 and 7 will not intersect to form a triangle. Sad.

Imagine the bottom angles’ vertices are hinges in the above example as well. And the sides of length 5 and 10 can pivot on those hinges. Because the sum of 5 and 10 is equal to 15, the sides of 5 and 10 will intersect, but they will intersect only after they are collapse to the bottom segment. This is not a triangle. Sad.

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Try the following examples. If you hold the left click of your mouse down and slide the mouse beside the problems, you will see the answers.

Can the following lengths be those of the three sides of a triangle?
1. 9, 10, 15  Yes, 9 + 10 = 19 which of course is greater than 15.
2. 3, 4, 5  Yes, 3 + 4 > 5.
3. 3, 4, 7   No, 3 + 4 is not greater than 7. Being equal is not good enough. Sorry.
4. 3, 4, 10  No, 3 + 4 is not greater than 10.

5. 7, 7, 10  Here we have an isosceles example. 7 + 7 > 14, yes this is a triangle.
6. 7, 7, 20  7 + 7 is not greater than 20, no triangle.
7. 2, 7, 7  What do we do here? Since the 7s are the larger, this will always be a triangle. The rule is that any two sides’ sum must be greater than the third side. This will always be true.
8. 10, 10, 10  Here we have an equilateral example. These will always be possible. Yes this is a triangle.

Geometry Notes — Midsegment of a Triangle

Midsegment of a Triangle
A line segment joining the midpoints of two sides of a triangle. A triangle has 3 possible midsegments.

Properties
1. The midsegment is always parallel to the third side of the triangle.
2. The midsegment is always half the length of the third side.
3. A triangle has three possible midsegments.

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Triangle Midsegment Theorem
If a segment joins the midpoints of two sides of a triangle, then the segment is parallel to the third side, and is half its length.

Geometry Assignment — Handout 5210

Geometry Assignment — Handout 5310

Geometry Biography — Leonhard Paul Euler

Leonhard Paul Euler (April 15, 1707 – September 18, 1783) was a pioneering Swiss mathematician and physicist who spent most of his life in Russia and Germany. Euler made important discoveries in fields as diverse as calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, such as the notion of a mathematical function. He is also renowned for his work in mechanics, optics, and astronomy. Euler is considered to be the preeminent mathematician of the 18th century and one of the greatest of all time.

Euler was born in Switzerland, in the town of Basel to Paul Euler, a pastor of the Reformed Church, and Marguerite Brucker, a pastor’s daughter. Paul Euler’s friend Johann Bernoulli, who was regarded as Europe’s foremost mathematician, would eventually be the most important influence on young Leonhard. At the age of thirteen Euler attended the University of Basel, and in 1723, received a master’s degree with a dissertation that compared the philosophies of Descartes and Newton. At this time, he was receiving Saturday afternoon lessons from Johann Bernoulli, who quickly discovered his new pupil’s incredible talent for mathematics.

Euler was at this point studying theology, Greek, and Hebrew in order to become a pastor. Johann Bernoulli convinced Paul Euler that Leonhard was destined to become a great mathematician. In 1726, Euler completed his Ph.D. In 1727, he entered the Paris Academy Prize Problem competition, where the problem that year was to find the best way to place the masts on a ship. He won second place, losing to Pierre Bouguer, who is now known as “the father of naval architecture”. Euler subsequently won this coveted annual prize twelve times in his career.

Euler arrived in the Russian capital on May 17, 1727. He was promoted to a position in the mathematics department at the Imperial Russian Academy of Sciences in St Petersburg. Euler mastered Russian and settled into life in St Petersburg. He also took on an additional job as a medic in the Russian Navy. The Academy at St. Petersburg, established by Peter the Great, was intended to improve education in Russia and to close the scientific gap with Western Europe. It was especially attractive to foreign scholars like Euler. The academy possessed ample financial resources and a comprehensive library drawn from private libraries of Peter himself and of the nobility. Very few students were enrolled in the academy so as to lessen the faculty’s teaching burden, and the academy emphasized research and offered its faculty the time and the freedom to pursue scientific questions.

On January 7, 1734, Euler married Katharina Gsell. The young couple bought a house by the Neva River. Of their thirteen children, only five survived childhood. SAD. Concerned about turmoil in Russia, Euler left St. Petersburg on June 19, 1741 to take up a post at the Berlin Academy, which he had been offered by Frederick the Great of Prussia. In addition, Euler was asked to tutor the Princess of Anhalt-Dessau, Frederick’s niece. Euler wrote over 200 letters to her, which was later compiled into a best-selling volume entitled Letters of Euler on different Subjects in Natural Philosophy Addressed to a German Princess. This work contained Euler’s exposition on various subjects pertaining to physics and mathematics, as well as offering valuable insights into Euler’s personality and religious beliefs. This book became more widely read than any of his mathematical works, and it was published across Europe and in the United States. Its popularity testifies to Euler’s ability to communicate scientific matters effectively to a lay audience, a rare ability for a dedicated research scientist.

Despite Euler’s immense contribution to the Academy’s prestige, he was eventually forced to leave Berlin. This was partly because of a personality conflict with Frederick, who came to regard Euler as unsophisticated, especially in comparison to the circle of philosophers the German king brought to the Academy. Voltaire was among those in Frederick’s employ, and the Frenchman enjoyed a prominent position in the king’s social circle. Euler, a simple religious man and a hard worker, was very conventional in his beliefs and tastes. He was in many ways the direct opposite of Voltaire. [François-Marie Arouet (1694 - 1778), better known by the pen name Voltaire, was a French Enlightenment writer and philosopher known for his wit and defense of civil liberties including freedom of religion. He was an outspoken supporter of social reform. He frequently use of his works to criticize Catholic Church dogma and the French institutions of his day. Voltaire was one of several Enlightenment figures (along with John Locke - not from LOST - and Thomas Hobbes) whose works and ideas influenced important thinkers of both the American and French Revolutions.]

Euler’s eyesight worsened throughout his mathematical career. Three years after suffering a near-fatal fever in 1735 he became nearly blind in his right eye, but Euler rather blamed his condition on the painstaking work on cartography he performed for the St. Petersburg Academy. Euler’s sight in that eye worsened throughout his stay in Germany, so much so that Frederick referred to him as “Cyclops”. Euler later suffered a cataract in his good left eye, rendering him almost totally blind. Even so, his condition appeared to have little effect on his productivity, as he compensated for it with his mental calculation skills and photographic memory. Euler’s powers of memory and concentration were legendary. For example, Euler could repeat the Aeneid of Virgil. The Aeneid is a Latin epic poem written in the 1st century BC (between 29 and 19 BC) that tells the legendary story of Aeneas, a Trojan who traveled to Italy, where he became the ancestor of the Romans. It is written in dactylic hexameter. The first six of the poem’s twelve books tell the story of Aeneas’ wanderings from Troy to Italy; second half tells of the Trojans’ ultimately victorious war upon the Latins. With the aid of his scribes, Euler’s productivity on many areas of study actually increased. He produced on average one mathematical paper every week in the year 1775.

The situation in Russia had improved greatly, and in 1766 Euler accepted an invitation to return to the St. Petersburg Academy and spent the rest of his life in Russia. His second stay in the country was marred by tragedy. A fire in St. Petersburg in 1771 cost him his home, and almost his life. In 1773, he lost his wife of 40 years. Three years after his wife’s death Euler married her half sister. This marriage would last until his death.

On September 18, 1783, Euler passed away in St. Petersburg after suffering a brain hemorrhage. His eulogy was written by the French mathematician and philosopher Marquis de Condorcet. Condorcet commented, “…il cessa de calculer et de vivre – … he ceased to calculate and to live.” Condorcet told the story of two of Euler’s students who had independently summed seventeen terms of a complicated infinite series, only to disagree in the fiftieth decimal place; Euler settled the dispute by recomputing the sum in his head. Outstanding

Geometry Notes — The Euler Line

In the 18th century, the Swiss mathematician Leonhard Euler noticed that three of the many centers of a triangle are always collinear, that is, they always lie on a straight line. This line has come to be named after him – the Euler line. The three centers that have this surprising property are the triangle’s centroid , circumcenter and orthocenter.

In the figure, the centroid is the black middle point on the line. The centroid is the point where the three medians converge.

The circumcenter is the purple point. The circumcenter is the point where the perpendicular bisectors of the triangle’s sides converge.

The orthocenter is the red point. The orthocenter is the point where the three altitudes of the triangle converge.

The Euler Line