Geometrical ratios

Question

In ΔABC, /AB/=10cm and /AC/ = 18cm. M, the midpoint of B¯C and the bisector of A meets BC at N. If MN=2cm. Calculate BC

Answer 1

Hi Damilola, you can solve this by using the angle-bisector theorem. The angle-bisector theorem is a proportion based on the sides of the triangle.

 

Given: AB = 10cm, AC = 18cm, point M is the midpoint of BC and the bisector of angle A meets BC at N. Additionally, The length of MN is 2cm.

 

If we let a represent the length of side AB, b represent the length of side AC, c represent the length of the line where N meets BC from the B angle, and last d represent the length of the line from N to angle C, we can the necessary proportion:

 

a / b = c / d

 

Points N is not the midpoint of BC, because if it was it would equal M. That means BC is split into two parts, one part has length x + 2 and the other part has length x - 2.

 

Using this information, substitute for a, b, c, and d:

 

a = 10cm

b = 18cm

c = x - 2

d = x + 2

 

Proportion:

 

10 / 18 = (x - 2) / (x + 2)

 

Cross-multiply:

 

10 * (x + 2) = 18 * (x - 2)

 

Clear parenthesis:

 

10x + 20 = 18x - 36

 

Subtract 10x from both sides and add 36 to both sides:

 

56 = 8x

 

Divide both sides by 8:

 

x = 7

 

x represent 1 / 2 of BC, therefore, BC will have a length of 14cm.

 

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