Pre-Calculus Notes — Proof for Heron’s Formula

Geometry Notes — Special Right Triangles — 45-45-90 and 30-60-90

You know this… A triangle with one right angle is called a right triangle. The side opposite the right angle is called the hypotenuse of the triangle. The other two sides are called legs. The other two angles are always acute and complementary.

The triangle above has side c as its hypotenuse, sides a and b as its legs. Angle C is its right angle. Angles A and B are acute and complementary.

There are two types of right triangles that every geometry student should know very well.

One is the right triangle formed when an altitude is drawn from a vertex of an equilateral triangle, forming two congruent right triangles.
The angles of the triangle will be 30, 60, and 90 degrees, giving the triangle its name: 30-60-90 triangle.
The ratio of side lengths in such triangles is always the same: if the leg opposite the 30 degree angle is of length x, the leg opposite the 60 degree angle will be of x, and the hypotenuse across from the right angle will be 2x.

The other common right triangle results from the pair of triangles created when a diagonal divides a square into two triangles.
Each of these triangles is congruent, and has angles of measures 45, 45, and 90 degrees.
If the legs opposite the 45 degree angles are of length x, the hypotenuse has a length of x. This ratio holds true for all 45-45-90 triangles. 45-45-90 triangles are also often called isosceles right triangle.

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45°- 45°- 90° Triangle – a right triangle where the angles are 45°, 45°, and 90°.

45-45-90 Triangle Theorem – In a 45-45-90 triangle, the hypotenuse is times as long as each leg.

The side-length ratios of leg: leg: hypotenuse are 1 : 1 : .

This is one of the ’standard’ triangles you should be able recognize on sight.

A fact you should commit to memory is: The sides are always in the ratio 1 : 1 : . With the ‘‘ being the hypotenuse (longest side). This can be derived from Pythagoras’ Theorem. This ratio will come in handy later in the study of trigonometry.

Because the base angles are the same (both 45°) the two legs are equal and so the triangle is also isosceles.

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30°- 60°- 90° Triangle – a right triangle where the angles are 30°, 60°, and 90°.

30-60-90 Triangle Theorem – In a 30-60-90 triangle, the hypotenuse is twice as long as the shorter leg, and the longer leg is times as long as the shorter leg.

The side-length ratios of short leg: long leg: hypotenuse are

This is one of the ’standard’ triangles you should be able recognize on sight. A fact you should commit to memory is: The sides are always in the ratio 1 : : 2. With the ‘2′ being the hypotenuse (longest side). This ratio will come in handy later in the study of trigonometry. Also notice that the smallest angle is always opposite the smallest side.

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The Wheel of Theodorus.

Theodorus of Cyrene was the tutor of Plato, teaching him mathematics. He was a member of the society of Pythagoras.

Statistics Assignment — Handout 5R20 and Key

Geometry Notes – Geometric Mean Examples

When the altitude (height) is drawn to the hypotenuse, it divides the hypotenuse in a way such that its length is the geometric mean between the two segments of the hypotenuse.

h is the altitude, x and y are the segments of the hypotenuse

Geometry Notes – Similar Right Triangles and Geometric Mean

There’s a lot going on below. ∆ABC is a right triangle. ∠C is the right angle. AB is the hypotenuse. CD is an altitude drawn to the hypotenuse. AD and BD are segments of the hypotenuse.

Theorem: If the altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are similar to the original triangle and to each other.

In this example, the angles in the small triangle ∆ACD are congruent to the angles in the medium triangle ∆CBD and are congruent to the angles in the large triangle ∆ABC.

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Theorem: In a right triangle, the length of the altitude from the right angle to the hypotenuse is the geometric mean of the lengths of the two segments of the hypotenuse.

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Take a look at the following drawing. Notice the highlighted boxes. equals .

This is a classic geometric mean setup. Anytime, in a proportion when a number or unknown can be cross-multiplied with itself, this is a geometric mean setup.

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Theorem: In a right triangle with an altitude drawn to the hypotenuse, the length of each leg is the geometric mean of the hypotenuse and the segment of the hypotenuse that is adjacent to that leg.

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Take a look at the following drawing. Notice the highlighted boxes.  and . These are classic geometric mean setups.

Statistics Assignment — Handout 5R10

Geometry — Box Bonus

I have a box that is 9 inches tall, 14 inches wide and 18 inches long. Find the length of the diagonal that passes through the interior of the box connecting point A and point B. See the pretty drawing.

The Art of War

SUN TZU ON THE ART OF WAR Chapter 6

Sun Tzu said: Whoever is first in the field and awaits the coming of the enemy, will be fresh for the fight; whoever is second in the field and has to hasten to battle will arrive exhausted.

Geometry Notes – Pythagorean Theorem and its Converse

“Silence is better than unmeaning words.” Pythagoras

The Pythagorean Theorem: In a right triangle, the length of the hypotenuse squared is equal to the sum of the squares of the lengths of the two legs.

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The equation for the Pythagorean Theorem is a very useful formula. Of course we can use it to find any missing side of a right triangle. If that wasn’t amazing enough, it is also a mechanism that allows us to classify a triangle according to its angles using the length of the three sides.

So for instance, if we have a triangle that has the lengths 6, 8, and 10 inches, we can plug these numbers into the Pythagorean theorem to verify if these lengths belong to a right triangle. And if they don’t, then we can, based on the results, state whether the triangle is acute or obtuse. Outstanding.

So..

6, 8, and 10 are lengths that belong to a right triangle.

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The Converse of the Pythagorean Theorem: In a triangle, if the square of the length of the largest side is equal to the sum of the squares of the lengths of the two smaller sides, then the triangle is a right triangle.

Notice that the length of the larger side, c, squared is equal to the sum of the squares of the length of the smaller sides, a and b.

We don’t assume that c is the hypotenuse, since we didn’t know we were working with a right triangle, but now we know c is a hypotenuse of a right triangle.

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The Acute Triangle Test: In a triangle, if the square of the length of the largest side is less than the sum of the squares of the lengths of the two smaller sides, then the triangle is an acute triangle.

Notice that the length of the larger side, c, squared is not equal to the sum of the squares of the length of the smaller sides, a and b. Its less. This is an acute triangle.

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The Obtuse Triangle Test: In a triangle, if the square of the length of the largest side is greater than the sum of the squares of the lengths of the two smaller sides, then the triangle is an obtuse triangle.

Notice that the length of the larger side, c, squared is not equal to the sum of the squares of the length of the smaller sides, a and b. Its more. This is an obtuse triangle.

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The Pythagorean Theorem: In a right triangle, the length of the hypotenuse squared is equal to the sum of the squares of the lengths of the two legs.

The Converse of the Pythagorean Theorem: In a triangle, if the square of the length of the largest side is equal to the sum of the squares of the lengths of the two smaller sides, then the triangle is a right triangle.

3 Triangle Tests:

Statistics — Biography — Abraham de Moivre

Abraham de Moivre (26 May 1667 in Vitry-le-François, Champagne, France – 27 November 1754 in London, England) was a French mathematician famous for de Moivre’s formula, which links complex numbers and trigonometry, and for his work on the normal distribution and probability theory.

Because of problems related to nationality and religion, Abraham de Moivre never had an opportunity to teach mathematics at a university. Nonetheless, he enjoyed fruitful interactions with Sir Isaac Newton and others and later published texts in which he advanced the understanding of probability theory and other areas of mathematics.

De Moivre was the son of a surgeon. His family belonged to the Huguenot sect, a Protestant group protected since 1598 by the Edict of Nantes, which ensured their limited freedom within the Catholic nation. Thus, de Moivre was educated in both Catholic and Protestant schools before going to Paris, where he studied under the renowned teacher Jacques Ozanam. When de Moivre was in his 20s, however, the Crown revoked the Edict of Nantes, changing the course of his career. After a three-year imprisonment, he fled to England, where he would spend the remainder of his life.

Though his foreign ancestry made it difficult for him to obtain employment through most recognized channels, de Moivre made a living by providing mathematical advice to gamblers and underwriters, who visited him at Slaughter’s Coffee House on Fleet Street in London. Newton’s support further ensured a steady stream of tutorial students. In 1695 another influential friend, Edmond Halley (1656-1742), presented a paper by de Moivre before the Royal Society, which accepted him for membership two years later. During the next half-century, de Moivre would have some 15 papers published in the Philosophical Transactions of the Royal Society.

As a mathematician, de Moivre concerned himself with areas ranging from probability theory to calculus. Among his contributions to mathematics was de Moivre’s theorem, which helped establish the close connection between the algebra and geometry of complex numbers. His writings on probability, translated into English as Doctrine of Chances (1718), would prove highly popular and highly influential. Ee Moivre wrote in an easy, understandable style geared toward nonmathematicians. This book, in which he improved on ideas presented earlier by mathematicians such as Jakob Bernoulli (1654-1705), is often referred to as the first modern probability textbook.

De Moivre was a Calvinist. Calvinism is a theological system and an approach to the Christian life that emphasizes the rule of God over all things. The purpose of de Moivre’s work was one of pressing concern in the Enlightenment, as he and other men of faith attempted to establish a rational basis for a belief in God. Thus, he hoped to use mathematics to prove the so-called “Argument from Design,” which maintains that the evidence of order in the universe proves the existence of God.

In 1735 de Moivre was honored with membership in the Berlin Academy and, almost two decades later, by something much more surprising: membership in the Paris Académie in the country that had once expelled him. By then, de Moivre was in his last days and he died in London on November 27, 1754.