Equation - Perpendicular Lines

Category: Analytic Geometry, Algebra

"Published in Newark, California, USA"

Find the equation of a line that passes through the point (2, -1) and perpendicular to the line 2x - 3y + 4 = 0.

Solution:

Consider the given line

Rewrite the equation of a line in slope-intercept form as follows

The slope of a line is m₁ = ²/₃ and the y-intercept is b = ⁴/₃. To draw or sketch a line, plot  ⁴/₃ at the y-axis. This is your first point of the line at (0, ⁴/₃). Next, use the slope to get the second point. From the first point, count 3 units to the right and then 2 units upward. Connect the two points and you have now a line as follows

Photo by Math Principles in Everyday Life

Finally, plot (2, -1) and draw a line perpendicular to the given line that contains a given point as follows

Photo by Math Principles in Everyday Life

If two lines are perpendicular, then their slopes are negative reciprocals to each other which means that m₂ = -¹/_m1 = ^-3/₂. Therefore, using the point-slope form, the equation of another line that passes through the point (2, -1) is

Equation - Parallel Lines

Category: Analytic Geometry, Algebra

"Published in Newark, California, USA"

Find the equation of a line that passes through the point (2, -1) and parallel to the line 2x - 3y + 4 = 0.

Solution:

Consider the given line

Rewrite the equation of a line in slope-intercept form as follows

The slope of a line is m₁ = ²/₃ and the y-intercept is b = ⁴/₃. To draw or sketch a line, plot  ⁴/₃ at the y-axis. This is your first point of the line at (0, ⁴/₃). Next, use the slope to get the second point. From the first point, count 3 units to the right and then 2 units upward. Connect the two points and you have now a line as follows

Photo by Math Principles in Everyday Life

Finally, plot (2, -1) and draw a line parallel to the given line that contains a given point as follows

Photo by Math Principles in Everyday Life

If two lines are parallel, then their slopes are equal, which is m₁ = m₂ = ²/₃. Therefore, using the point-slope form, the equation of another line that passes through the point (2, -1) is

More Spherical Triangle Problems

Category: Trigonometry

"Published in Newark, California, USA"

Find the side opposite the given angle for a spherical triangle having

(a) b = 60°, c = 30°, A = 45°
(b) a = 45°, c = 30°, B = 120°

Solution:

A spherical triangle is a triangle whose sides are the edges of a sphere. It is not the same as a plane triangle because the sides of a spherical triangles are curve and not a straight line. A spherical triangle looks like this

Photo by Math Principles in Everyday Life

As you notice that the measurements of the edges of a spherical triangle are expressed in degrees because the sides of a spherical triangle are the arcs of a sphere. The measurements of the arcs of a sphere or the edges of a spherical triangle are measured from the center of a sphere. 

In this case, we will use the formulas that are completely different from the formulas of plane triangles. The formulas that we will use are the following:

The above formulas are called the Law of Cosines. You must  remember or memorize all the above formulas because you will use these often in solving the sides and the angles of spherical triangles.

Now, let's go back to the given problem, if 

(a) b = 60°, c = 30°, A = 45°

then we have to solve for the measurement of arc a. Use the first formula as follows

Substitute the values b, c, and A, we have

or

Let's have another one, if

(b) a = 45°, c = 30°, B = 120°

then we have to solve for the measurement of arc b. Use the second formula as follows

or