Chapter 5 Section 5.6 Study Guide

Weijia Wang, GEOMETRY II

Chapter 5 section 5.6 Study Guide:

The Distance Formula:

Solve for: The distance between point A and point C. Length of AC.

As you can see, line segment AC is neither horizontal nor vertical, but diagonal. If you drop down a line from point A and move point C over horizontally, where the two rays intersect (point B) is the right angle of the newly formed right triangle. After the triangle is formed, solving the distance becomes much clearer and easier – just use the Pythagorean Theorem.

·  Length of leg AB is 4 – difference of the y-value for point A and point C.

·  Length of leg BC is 3 – difference of the x-value for point A and point C.

·  Then all you have to do to solve for the distance of AC is plug the length of AB and BC into the equation: d = √(x2 + y2)

·  Distance between point A and point C is 5.

If you only plugged in the variables, this would be the equation:

Quadrature:

(Left-hand rule) (Right-hand rule)

When you’re trying to estimate the area of one part of a circle on a graph such as the one shown above, you can use quadrature – using left-hand rule or right-hand rule.

·  When you use the left-hand rule, the estimation will be larger than the actual area, so the upper-left vertex of each rectangle will intersect the circle, hence the name left-hand rule.

·  When you use the right-hand rule, the estimation will be less than the actual area, so the upper-right vertex of each rectangle will intersect the circle, hence the name right-hand rule.

To find the estimated area, you first need to find the area of each rectangle with a base of 1.

·  To find the area of each rectangle, you need the height, since height * 1(base) = area

·  To find the height of each rectangle, use the Pythagorean Theorem.

·  The left-hand rule diagram clearly shows each height.

·  The hypotenuse will always stay the same because it is the radius of the circle.

·  One leg will be multiples of 1 depending what the x-value of the point is.

·  Then use the Pythagorean Theorem to find the missing leg, which is the height.

·  The total of the area of each rectangle is the estimate of that part of the circle.

(Right-hand rule is calculated the same way as left-hand rule)

Combining Method:

·  This will just give you a better estimated area compared to if you only used one of the method.

·  To use this method, just find the average of the left-hand rule and the right-hand rule of the same circle.