# Pythagorean theorem

**Problem**

A circle has an equation of $x^2 + y^2 + 2cy = 0$. Find the value of $c$ when the length of the tangent from (5, 4) to the circle is equal to one.

A. 5 | C. 3 |

B. -3 | D. -5 |

## 10 Solving for angle A in triangle ABC

**Problem 10**

In a triangle *ABC*, if $\dfrac{2 \cos A}{a} + \dfrac{\cos B}{b} + \dfrac{2 \cos C}{c} = \dfrac{a}{bc} + \dfrac{b}{ca}$, find the value of angle $A$.

## 26-27 Time Rates: Kite moving horizontally

## 11-12 Two trains; one going to east, and the other is heading north

**Problem 11**

A train starting at noon, travels north at 40 miles per hour. Another train starting from the same point at 2 PM travels east at 50 miles per hour. Find, to the nearest mile per hour, how fast the two trains are separating at 3 PM.

## 10 - A boy on a bike

**Problem 10**

A boy on a bike rides north 5 mi, then turns east (Fig. 47). If he rides 10 mi/hr, at what rate does his distance to the starting point S changing 2 hour after he left that point?

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## 06-07 Ladder slides down the wall

**Problem 06**

A ladder 20 ft long leans against a vertical wall. If the top slides downward at the rate of 2 ft/sec, find how fast the lower end is moving when it is 16 ft from the wall.

## Derivation of Pythagorean Identities

In reference to the right triangle shown and from the functions of a right triangle:

a/c = sin θ

b/c = cos θ

c/b = sec θ

c/a = csc θ

a/b = tan θ

b/a = cot θ

## Derivation of Pythagorean Theorem

**Pythagorean Theorem**

In any right triangle, the sum of the square of the two perpendicular sides is equal to the square of the longest side. For a right triangle with legs measures $a$ and $b$ and length of hypotenuse $c$, the theorem can be expressed in the form