Proving - Inscribed Triangle, Circle

Category: Plane Geometry

"Published in Newark, California, USA"

Given: ∆ABC inscribed in circle O

Prove: m∠BAC = ½ m(arc BC)

Photo by Math Principles in Everyday Life

Solution:

Consider the given figure and we will label further as follows

Photo by Math Principles in Everyday Life

Proof:

1. Statement: From point A, draw a line that passes through the center of a circle.

    Reason: Line AO will be used later to get m∠BOC.

2. Statement: Draw radius OB and let m∠BAO = x.

    Reason: Radius OB will be used to create an isosceles ∆BOA and x as the base angle.

3. Statement: OB ≅ OA and m∠BAO ≅ m∠OBA.

    Reason: OB and OA are the radius of a circle and the two sides of an isosceles triangles are congruent. The two base angles of an isosceles triangles are congruent.

4. Statement: m∠BOA = 180° - 2x

    Reason: The sum of the three angles of a triangle is always equal to 180°.

5. Statement: m∠BOD = 180° - (180° - 2x) = 2x

    Reason: The sum of supplementary angles is always 180°. m∠BOA and m∠BOD are supplementary angles.

6. Statement: Draw radius OC and let m∠CAO = y.

    Reason: Radius OC will be used to create an isosceles ∆COA and y as the base angle.

7. Statement: OC ≅ OA and m∠CAO ≅ m∠OCA.

    Reason: OC and OA are the radius of a circle and the two sides of an isosceles triangles are congruent. The two base angles of an isosceles triangles are congruent.

8. Statement: m∠COA = 180° - 2y

    Reason: The sum of the three angles of a triangle is always equal to 180°.

9. Statement: m∠COD = 180° - (180° - 2y) = 2y

    Reason: The sum of supplementary angles is always 180°. m∠COA and m∠COD are supplementary angles.

10. Statement: m∠BAC = x + y and m∠BOC = 2x + 2y.

      Reason: Addition property of angles

11. Statement: m∠BAC = ½ m∠BOC

      Reason: If OA, OB, and OC are equidistant from its center at O, then m∠BAC = ½ m∠BOC.

12. Statement: m∠BOC = m(arc BC)

      Reasons: If m∠BOC is a central angle formed by the two radii OB and OC, then m(arc BC) is an arc formed by a central angle. 

13. Statement: m∠BAC = ½ m∠BOC = ½ m(arc BC)

      Reasons: Substitution proposition.

Triangle Inscribed - Circle Problem

Category: Trigonometry, Plane Geometry

"Published in Newark, California, USA"

If R is the radius of a circle circumscribed about the triangle ABC, show that 

Photo by Math Principles in Everyday Life

Hint: ∠BAC = ∠DOC

Solution:

Consider ∆DOC

Consider ∆ABC and apply Sine Law, we have

but

Therefore,

Money - Investment Problem, 2

Category: Algebra

"Published in Newark, California, USA"

Aling Etang bought several dozen eggs at PHP 600. Because of poor storage, 2 dozens were spoiled and had to be thrown away. Nonetheless, she realized a profit of PHP 154 by charging PHP 3 more than she paid per dozen. How many dozen eggs did she buy?

Solution: 

The given word problem is about money and investment problem with profit, of course where she thrown 2 dozens of eggs but she earned a profit by charging more than she paid per dozen in order to save her capital or investment from loss. Let's analyze the problem as follows

Let x = be the price of a dozen of eggs, in PHP
      y = be the number of dozens of eggs

From the first word statement, "Aling Etang bought several dozen eggs at PHP 600", the working equation will be

From the next word statement, "Because of poor storage, 2 dozens were spoiled and had to be thrown away", the working equation will be

From the next word statement, "....by charging PHP 3 more than she paid per dozen," the working equation will be

The overall working equation for this problem is

               Profit = Total Amount Sold - Total Amount Invested

but

and

The above equation becomes

Divide both sides of the equation by -2, we have

Multiply both sides of the equation by x, we have

Use Quadratic Formula to solve for the value of x, we have

To calculate the price of a dozen of eggs, choose the positive sign as follows

Therefore, the price of a dozen of eggs is PHP 10. 

Consider the first equation

if x = 10, then 

Therefore, Aling Etang bought 60 dozens of eggs. 

Note: PHP means Philippine Pesos and the prices mentioned in the word problem above was in 1985.