Math Is Fun Forum

Enshrouded_

Member

Registered: 2015-07-31

Posts: 47

Geometry

In triangle

, the medians

,

, and

concur at the centroid

(a) Prove that

.

(b) Let

be the perimeter of

Prove that

Hint(s):
(a) Connect

. What does the Triangle Inequality say about

?

(b) For one inequality, use part (a). For the other inequality, use the Triangle Inequality on triangles

,

, and

.

Last edited by Enshrouded_ (2015-09-01 13:36:19)

Bob

Administrator

Registered: 2010-06-20

Posts: 10,626

Re: Geometry

hi Enshrouded_

(a)  Start with AD < AE + ED

Remember how the medians are created.  (D, E and F are the midpoints so AE is half AC and DE is parallel to AB and half its length)

(b) Write the two similar inequalities for BE and CF.  Add the three together.  (If P < Q and R < S then P+Q < R+S)

Construct the three inequalities using ABG, BCG and CAG.  (eg. AB < AG + BG)

Add them up.

Remember the centroid is one third of the way up the median. (eg. AG = 2.AD/3)

Hope that's enough for you to complete these.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Enshrouded_

Member

Registered: 2015-07-31

Posts: 47

Re: Geometry

I got everything but the last step. How do I prove that AD + BE + CF<P

Bob

Administrator

Registered: 2010-06-20

Posts: 10,626

Re: Geometry

Write the two similar inequalities for BE and CF.  Add the three together.  (If P < Q and R < S then P+Q < R+S)

When you do this you should end up with AD + BE + CF < AB + BC + CA

It takes a couple of steps.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob