   Chapter 10.CT, Problem 20CT Elementary Geometry For College St...

7th Edition
Alexander + 2 others
ISBN: 9781337614085

Solutions

Chapter
Section Elementary Geometry For College St...

7th Edition
Alexander + 2 others
ISBN: 9781337614085
Textbook Problem

Use the drawing provided for the proof of the theorem “The line segment that joins the midpoints of two sides of a triangle is parallel to the third side of the triangle.” Complete this Proof:Proof: Given △ A B C with vertices as shown, let M and n name the midpoints of sides C B - and A B - , respectively.

To determine

To prove:

The given theorem “The line segment that joins the midpoints of two sides of a triangle is parallel to the third side of the triangle”.

Explanation

The given figure is shown below.

We have to prove the given theorem.

Let A0, 0, B2a, 0 and C2b, 2c be the vertices of the triangle.

Let M be the midpoint of the line segment CB-.

Let N be the midpoint of the line segment AB-.

The line segment that joins the midpoints is MN.

We have to prove that MN- is parallel to AC-

That is, slope of MN- is equal to slope of AC-.

Find the vertices of M.

Since, M lies midway between B and C.

Find the midpoint of -

This is the coordinate of M.

x1,y1=B2a, 0

x2,y2=C2b, 2c

Formula for midpoint of the line segment joining two points.

M=x1+x22, y1+y22

Substituting the x and y co-ordinates.

M=2a+2b2, 0+2c2

M=2a+b2, 2c2

Cancelling 2,

M=a+b, c

Find the vertices of N.

Since, N lies midway between A and B.

Find the midpoint of AB-.

This is the coordinate of N.

x1,y1=A0, 0

x2,y2=B2a, 0

Substituting the x and y co-ordinates.

N=0+2a2, 0+02

N=2a2, 02

Cancelling 2,

N=a, 0

Now, prove that MN-AC-

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