Free counters!

Friday, January 3, 2014

Circle - Triangle Problems

Category: Trigonometry, Plane Geometry

"Published in Vacaville, California, USA"

Express the lengths a and b in the figure in terms of the trigonometric ratios of θ

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 a circle and a triangle. One side of a triangle is equal to the radius of a circle which is 1. The other side of a triangle which is a is tangent to a circle at the y-axis. If a is perpendicular to the other side of a triangle and to the y-axis, then it follows that the given triangle is a right triangle.

Photo by Math Principles in Everyday Life
   
If a is perpendicular to y-axis, then it follows that a is parallel to x-axis. If b is a transversal line that passes thru the parallel lines a and the x-axis, then it follows that the alternating interior angles are congruent. In this case, the angle formed by lines b and the x-axis is congruent to the angle formed by lines a and b which is θ.

Since we know the side and the opposite angle of a right triangle, then we can get the values of a and b in terms of trigonometric functions of θ as follows






 

Thursday, January 2, 2014

More Triangle Problems, 2

Category: Trigonometry, Plane Geometry

"Published in Vacaville, California, USA"

Solve for the values of x and y for the given figure:

Photo by Math Principles in Everyday Life

Solution:

Consider the given figure above

Photo by Math Principles in Everyday Life

There are two right triangles that are adjacent to each other. The hypotenuse of a right triangle is equal to the side of the other right triangle. Since the values of one side and two angles are given, then we can solve for x and y using trigonometric functions as follows












Wednesday, January 1, 2014

Area - Triangle, Given Three Vertices

Category: Analytic Geometry, Plane Geometry

"Published in Vacaville, California, USA"

Refer to triangle ABC in the figure.

(a) Show that triangle ABC is a right triangle by using the converse of the Pythagorean Theorem.
(b) Find the area of triangle ABC.

Photo by Math Principles in Everyday Life

Solution:

Consider the given figure above. To illustrate further the problem, let's label further the given figure as follows

Photo by Math Principles in Everyday Life

(a) The first thing that we have to do is to get the length of each sides of a triangle using the distance of two points formula.

The length of AB is






The length of BC is







The length of AC is







Next, apply the Pythagorean Theorem by substituting the values of the sides of the triangle. Since AC is the longest side, then it is the hypotenuse of a right triangle. 






Since both sides of the equation are equal, then ΔABC is a right triangle.

(b) If the vertices of a triangle are given, then we can get the area of a triangle as follows









Since the area of any plane figure is always in absolute value or positive value, then the area of a triangle is