In the triangle ABC

Question

AB=4.2cm BC=5.6cm D is the midpoint of AC, BD=3.5cm 

Calculate the length of AC

Answer 1

The angle at vertex B is 90 degrees, which simplifies the solution if that was given. Using that information the side AC^2 = BC^2 + AB^2, so AC = sqrt(5.6^2 + 4.2^2) = sqrt(49) = 7 cm. 

 

You get the same answer if the above hint is not provided using the law of cosines. 

 

It helps to draw a picture to follow. Let angle alpha be the angle at vertex A. We have these two equations:

 

(1) 5.6^2 = 4.2^2 + AC^2 -2*(4.2)*(AC)*cos(alpha)

 

(2) 3.5^2 = 4.2^2 + (AC/2)^2 -2*4.2*(AC/2)*cos(alpha) 

 

Multiply (2) by -2 and add, giving this equation:

 

5.6^2 - 2*3.5^2 = -4.2^2 +.5*(AC^2)

 

so

AC^2 = 2*(5.6^2 + 4.2^2 -2*3.5^2) = 49

 

so 

AC = 7, same answer.

 

If you calculate the angles that shows the angle at B is 90 degrees.

Answer 2

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