More Triangle Problems, 3

Category: Trigonometry, Plane Geometry

"Published in Vacaville, California, USA"

For the triangle shown, find ∠BCD and ∠DCA.

Photo by Math Principles in Everyday Life

Solution:

Consider the given figure above

Photo by Math Principles in Everyday Life

The unknown angles of two adjacent triangles can be solved by using Sine Law.

Consider ΔABC:

Use Sine Law in order to solve for ∠ABC, we have

 

 

 

 

 

 

Consider ΔBCD:

Since BC ≅ CD = 20, then it follows that ΔBCD is an isosceles triangle. If the two sides of an isosceles triangle are equal, then the two base angles are equal also. In this case,
∠DBC ≅ ∠CDB = 44.427°. Therefore,

 

 

Consider ΔACD:

Since ∠BDC and ∠ADC are supplementary angles, then we can solve for ∠ADC as follows

 

 

Therefore,

 

 

Circular Segment Problems

Category: Plane Geometry, Trigonometry

"Published in Vacaville, California, USA"

Find the area of the shaded region in the figure:

Photo by Math Principles in Everyday Life

Solution:

Consider the given figure above

Photo by Math Principles in Everyday Life

The area of a circular segment is the difference of the area of circular sector and a triangle. 

The area of a circular sector is given by the formula

Substitute the values of r and θ, we have

Since the unit of area is square units, then the unit of an angle must be in radians since it is a unit less value of angle. Let's convert the unit of an angle in radians as follows

The area of a triangle given the two sides and the adjacent angle is given by the formula

Since the triangle is inscribed in a circle whose vertex is a center of a circle, then it follows that a = b = r. The above equation becomes

  

Substitute the values of r and θ, we have

Therefore, the area of circular segment is

Area - Triangle, Given Three Vertices, 4

Category: Analytic Geometry, Plane Geometry

"Published in Newark, California, USA"

If the area of a triangle with vertices (5, 2), (x, 4), and (0, -3) is 12 ½, find x.

Solution:

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

Photo by Math Principles in Everyday Life

The area of a triangle is given by the formula

Substitute the values of the coordinates of the vertices as well as the area, we have

Therefore, x = 2 in which the other vertex of a triangle is (2, 4).