Regular Polygon Problems, 3

Category: Plane Geometry

"Published in Newark, California, USA"

Find the area of a regular hexagon with perimeter 12 cm.

Solution:

To illustrate the problem, it is better to draw the figure as follows

Photo by Math Principles in Everyday Life

There are six triangles of a regular hexagon if you draw the three largest diagonal that passes through its center which is the vertex of each triangles. The first thing that we need to do is to get the vertex angle of a triangle in a regular hexagon as follows

If the given figure is a regular hexagon, then all vertex angles of the triangles are congruent. Also, the two adjacent sides of each triangle are congruent. If that's the case, the other two angles of a triangle are also congruent. The other two equal angles of a triangle are

Since all angles of a triangle are congruent, then the triangle is equilateral or equiangular triangle. 

If the perimeter of a regular hexagon is given, then we can solve for the length of a base of each equilateral triangle as follows

If you draw a perpendicular line segment from the vertex to the base of a triangle, then that line segment is the altitude of each triangle or the apothem of a regular hexagon. The altitude of a triangle bisects the base. There are two 30° - 60° right triangles of an equilateral triangle. By Pythagorean Theorem, the altitude of a triangle is

Therefore, the area of a regular hexagon is

                            or

 

Equilateral Triangle Problems

Category: Plane Geometry

"Published in Newark, California, USA"

Find the area of an equilateral triangle inscribed in a circle with radius 2√3.

Solution:

To illustrate the problem, it is better to draw the figure as follows

Photo by Math Principles in Everyday Life

If an equilateral triangle is inscribed in a circle, then the radius of a circle bisects the angles of an equilateral triangle. Each angle of an equilateral triangle is 60°. In this case if you analyze further the figure, there are six 30° - 60° right triangles. In this problem, we need the length of the base and the altitude of an equilateral triangle.

The base of an equilateral triangle is
 

 

The altitude of an equilateral triangle is
 

 

Therefore, the area of an equilateral triangle is
 

 

 

There's another way in getting the area of an equilateral triangle. If you know the sides of an equilateral triangle, then we can use the Heron's Formula as follows
 

The semi-perimeter of an equilateral triangle is
 

 

 

Therefore, the area of an equilateral triangle is
 

 

 

 

 

Circle and Secant Segment Problems

Category: Plane Geometry

"Published in Newark, California, USA"

Find the values of x and y in the given figure

Photo by Math Principles in Everyday Life

Solution:

Consider the given figure above

Photo by Math Principles in Everyday Life

The given figure consists of two secant segments that are drawn in a circle from an external point. The other line with the same external point is tangent to a circle. If a theorem says "When two secant segments are drawn to a circle from an external point, the product of one secant segment and its external segment equals the product of the other secant segment and its external segment.", then we can solve for the values of x and y which are the secant and external segments.

By using the line segments AO and CO, the value of x is

By using the line segments OA and OE, the value of y is

 

 

Circle and Inscribed Angle Problems

Category: Plane Geometry

"Published in Newark, California, USA"

Find the values of x and y in the given figure

Photo by Math Principles in Everyday Life

Solution:

The given figure consists of two inscribed angles. Inscribed angle is an angle of which the vertex is located along the arc of a circle. The sides of the inscribed angles are the chords of a circle. ∠PSR and ∠PTQ are inscribed angles of circle O.

Let's consider ∠PTQ and draw the line segments OP and OQ as follows
 

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If a theorem says "The measure of an inscribed angle is equal to half the measure of its intercepted arc.", then the value of x which is the measure of ∠PTQ is
 

 

 

Let's consider ∠PSR and draw the line segments OP and OR as follows

Photo by Math Principles in Everyday Life

If a theorem says "The measure of an inscribed angle is equal to half the measure of its intercepted arc.", then the value of y which is the measure of arc QR is