Mathematics Magazine for Grades 1-12  

Grade 10

Theory:

Carnot’s Theorem .  In any triangle ABC the algebraic sum of the distances from the circumcenter O to the sides, is R+r, the sum of circumradius and the inradius.

Solutions from the Previous Issue:

1.        What is the maximum number for the possible points of intersection of a circle and a triangle?

Solution:

There are 6 possible points of intersection of a circle and a triangle.

2.        Solve for x:

Solution:

The procedure is to multiply both sides of the equation by the LCD, which in this case is (x – 2)(x + 1). It is essential to multiply every term on both sides of the equation. In so doing, all denominator cancel.

x + 1 –2(x – 2) = 7

x + 1 – 2x + 4 = 7

-x = 2

x = -2

3.        Solve for x:

Solution:

The LCD is (x – 3)((x – 2).

Multiplying both sides by this yields

3(x - 2) – 2(x – 3) = 3

3x – 6 – 2x + 6 = 3

x = 3

4.        Solve for x:

Solution:

We multiply both sides by the LCD, (x – 1)(x + 3), to obtain

x + 3 + 3(x – 1) = (x – 1)(x + 3)

x + 3 + 3x – 3 = x² + 2x –3

0 = x² - 2x – 3

(x –3)(x + 1) =0

x = 3; x = -1

Since the LCD is nonzero for each of these, the solution set is {-1,3}

Proposed Exercises 

Solve each system of equations

1.        16x - y = 283
3x + y = 59

2.        2x + 9y = -219
3x - 19y = 289

3.        8x + 13y = 44
9x + 13y = 82

4.        15x + 17y = 327
11x - 12y = -641

Read more on the written version of the publication.