12 - 14 Rectangular Lot Problems in Maxima and Minima

Problem 12
A rectangular field of fixed area is to be enclosed and divided into three lots by parallels to one of the sides. What should be the relative dimensions of the field to make the amount of fencing minimum?

09 - 11 Rectangular Lot Problems in Maxima and Minima

Problem 9
What should be the shape of a rectangular field of a given area, if it is to be enclosed by the least amount of fencing?
 

Area of Regular Six-Pointed Star

Problem
Find the area of the regular six-pointed star inscribed in a circle of radius 20 cm.
 

Circle Tangent Internally to Another Circle

Problem
Two circles as shown below are tangent to each other at point C. If AB = 9 cm and DE = FG = 5 cm, find the area of the shaded region.
 

Two Internally Tangent Circles

 

Area of Regular Five-Pointed Star

Problem
Find the area of the regular five-pointed star inscribed in a circle of radius 20 cm.
 

Relationship Between Central Angle and Inscribed Angle

The Central Angle Theorem on Circles | Geometry

Central angle = Angle subtended by an arc of the circle from the center of the circle.
Inscribed angle = Angle subtended by an arc of the circle from any point on the circumference of the circle. Also called circumferential angle and peripheral angle.
 

Figure below shows a central angle and inscribed angle intercepting the same arc AB. The relationship between the two is given by
 

$\alpha = 2\theta \, \text{ or } \, \theta = \frac{1}{2}\alpha$

 

if and only if both angles intercepted the same arc. In the figure below, θ and α intercepted the same arc AB.
 

The Circle

The following are short descriptions of the circle shown below.

Tangent - is a line that would pass through one point on the circle.
Secant - is a line that would pass through two points on the circle.
Chord - is a secant that would terminate on the circle itself.
Diameter, d - is a chord that passes through the center of the circle.
Radius, r - is one-half of the diameter.

 

Derivation of Formula for Radius of Circumcircle

The formula for the radius of the circle circumscribed about a triangle (circumcircle) is given by
 

$R = \dfrac{abc}{4A_t}$

where At is the area of the inscribed triangle.
 

Derivation of Formula for Radius of Incircle

The radius of incircle is given by the formula
 

$r = \dfrac{A_t}{s}$

 

where At = area of the triangle and s = semi-perimeter.
 

Centers of a Triangle

This page will define the following: incenter, circumcenter, orthocenter, centroid, and Euler line.
 

Incenter
Incenter is the center of the inscribed circle (incircle) of the triangle, it is the point of intersection of the angle bisectors of the triangle.
 

incenter-incircle.jpg

 

The radius of incircle is given by the formula

$r = \dfrac{A_t}{s}$

where At = area of the triangle and s = ½ (a + b + c). See the derivation of formula for radius of incircle.
 

Pages

Subscribe to MATHalino RSS