The Byzantine Empire **contributed the Hagia Sophia to European culture**. The Byzantine Empire was heavily influenced by Greek culture. Greek was the official language of the empire, and scholars in Byzantium preserved great Greek works that would later influence the Renaissance.

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How did Greek Fire help the Byzantine military? **They could burn ships even if they were in water**. How does a moat work? It surrounds a city with water to protect it.

How did religious practices differ in the Byzantine and Western Roman empires? In the Byzantine Empire, the highest political authority and highest religious authority was the same person. However, in the West, kings and church leaders were separate, though kings answered to the pope.

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Byzantine architecture was mostly influenced by **Roman and Greek architecture**. It began with Constantine the Great when he rebuilt the city of Byzantium and named it Constantinople and continued with his building of churches and the forum of Constantine.

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**Flamethrowers, hand grenades, portable sundials, musical organs, hydraulics, water cisterns, ship mills, and the fork** were among the many inventions of the Byzantines.

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The most important legacy of the Byzantine Empire is the **preservation of Greek and Roman civilization during the Middle Ages**. Byzantine civilization blended Christian religious beliefs with Greek science, philosophy, arts, and literature. They also extended Roman achievements in engineering and law.

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right-angled triangles

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The Pythagorean Theorem (its converse, really) **can be used on any triangle** to tell us whether or not it is a right triangle.

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One of the best known mathematical formulas is Pythagorean Theorem, which provides us with the relationship between the sides in a **right triangle**. A right triangle consists of two legs and a hypotenuse.

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This is a right triangle; when you sum the squares of the lengths of the sides, you get the square of the length of the hypotenuse. Incorrect. This is not a right triangle, so you cannot use the Pythagorean Theorem **to find r**.

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**The Pythagorean theorem states that the square of the hypotenuse, or longest side, of a right scalene triangle is equal to the sum of the squares of the two adjacent sides.**

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Q: Is sohcahtoa only for right triangles? A: **Yes, it only applies to right triangles**. If we have an oblique triangle, then we can’t assume these trig ratios will work. We have other methods we’ll learn about in Math Analysis and Trigonometry such as the laws of sines and cosines to handle those cases.

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**You can use the Pythagorean theorem and height of the right triangles within the equilateral to determine the missing side lengths of an equilateral triangle**. Then, you can use the formula A = √3/4 (a²) to determine the area of an equilateral triangle.

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As per the theorem, **the hypotenuse is the longest side of the triangle and is opposite the right angle**. Hence we can say that the Pythagorean theorem only works for right triangles.

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Sine Rule. **The Sine Rule can be used in any triangle (not just right-angled triangles) where a side and its opposite angle are known**. You will only ever need two parts of the Sine Rule formula, not all three. You will need to know at least one pair of a side with its opposite angle to use the Sine Rule.

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The mnemonic SOHCAHTOA can be used to aid in remembering which function to use in what circumstance ” **SOH stands for Sine is opposite over hypotenuse; CAH stands for Cosine is adjacent over hypotenuse; and TOA stands for Tangent is opposite over adjacent**. This will save confusion when working with these functions.

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Explanation: For Trigonometric functions to work you need a hypotenuse, which **you can only get in right triangles**. When you are dealing with triangles other than right triangles, the solution is to draw a perpendicular line to create right triangles.

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We can use SOHCAHTOA **to find a missing side of a right angled triangle when we have another side and a given angle**. We can use SOHCAHTOA to find a missing angle of a right angled triangle when we have two given sides.

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The Pythagorean theorem can be used to solve for any unknown side of a right triangle if the lengths of the other two sides are known. **The Pythagorean theorem can be used to solve for any side of an isosceles triangle as well, even though it is not a right triangle**.

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Formulas and Calculations for a equilateral triangle: Altitude of Equilateral Triangle h = (1/2) * √3 * a. Angles of Equilateral Triangle: A = B = C = 60° Sides of Equilateral Triangle: **a = b = c**.

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The easiest way is to **use a protractor to draw three 120° angles around the center of the circle, and then connect the three points on the circle where the three angles intersect the circumference**. That will result in an inscribed equilateral triangle. What is an ” apex” in an equilateral triangle?

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The hypotenuse formula is simply taking the Pythagorean theorem and solving for the hypotenuse, c . Solving for the hypotenuse, we simply take the square root of both sides of the equation a² + b² = c² and solve for c . When doing so, we get **c = √(a² + b²)** .

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