Midpoints of the sides

Question

Use Triangle GHJ where A, B, and C are midpoints of the sides.

 

If AB= 3x+8 adn GJ= 2x+24, what is AB?

 

If AC= 3y-5 amd HJ = 4y+2, what is HB? 

 

Id GH = 7z-1 and BC= 4z-3, what is GH? 

Answer 1

Hi Kim,

 

Without seeing the picture, I'm going to assume that:

A is the midpoint of GH

B is the midpoint of HJ

C is the midpoint of GJ

 

In a triangle, the segment connecting two midpoints is called a midsegment. A midsegment of a triangle is parallel to the the side that it does not connect, and is exactly half the length.

 

For your first pair of segments, you have AB = 3x+8 and GJ = 2x+24.

AB = 1/2 GJ or

2 AB = GJ

2 (3x+8) = 2x + 24  //Substituting the expressions for the segments

6x + 16 = 2x + 24   //distributive property on the left

4x = 8   //subtract 2x and 16 from both sides

x = 2  //divide both sides by 4

 

AB = 3x + 8

AB = 3(2) + 8  //substitute the solution for x

AB = 6+8 = 14.

 

You can solve the other two pairs of equations in the same way!

Answer 2

The segments that join the sides are ½ of the opposite sides.

 

So:  AB = ½GJ =>  3x + 8 = ½(2x + 24) => 3x + 8= x + 12 => 2x = 4 => x = 2.

 

So AB = 3(2) + 8 = 14.

 

AC = ½HJ and AC = HB.

 

Draw the triangle.  That will help.  This all boils down to the theorem that says that the segment joining the midpoints of a triangle is ½ of the opposite sides.