Thale's Theorem

Homework for Monday January 30, 2012

1)  ABC is an acute and scalene triangle.
     The circle of diameter [BC] cuts the sides [AB] and [AC] at E and F respectively.
     The circle of diameter [EF] cuts the sides [AB] and [AC] at M and N respectively.

      a)  Draw a figure.
      b)  Show that (BF) is parallel to (EN).
      c)  Show that the ratio AM / AF equals the ratio AE / AB and AM / AE equals AF / AC.
      d)  Deduce that AM x AC = AN x AB
      e)  Show that (MN) is parallel to (BC).


2)  ABCD is a parallelogram such that AB=8cm and AD=4.5cm.
      E is a point on [DA) such that DE=6cm. [EC) cuts [AB] at point I.

      a) Calculate AI,
      b) Construct point J on [DC) such that  DJ=three- fourth of DC.
      c) Show that (AJ) is parallel to (EC).

3)   BIEN is a parallelogram such that BI= 6cm and IE= 9cm and BE= 12cm.
      Let V be a point [BI] such that BV= 4cm.
     The parallel to [IE] passing through V cuts (BE) at D.

     a)  Show that BD= 8cm.
     b)  R is a point of [BN] such that BR= 6cm.
          Show that (DN) is parallel to (EN).
    c)  The parallel to (BE) passing through R cuts (IE)    at K and (NE) at H.
         Calculate HE and EK.

4)  ABC is a scalene triangle.
     a)  M is a point of [AC] such that the ratio CM o verCA =one- third. Construct point M.
     b)  Place point  D symmetric of B with respect to C.
          What does point M represent for triangle AB
     c) The parallel through M to (BD) cuts [AB] and [AD] at F and E respectively.
         Find   4 ratios equal to AM over AC.
         Deduce that M is the midpoint of [EF].

PUZZLE

It takes Hadi 6 hours to do a certain job, while it takes Rami 9 hours to do the same job.
If Hadi and Rami work together, how long would it take them to accomlish the same task?

answe as fast as possible.

Thale's Theorem

Our theme in this unit is Thale's Theorem.
The objectives of this unit are:
1) To calculate the lengths of segments by applying Thale's Theorem.
2) To prove that two lines are parallel by applying converse of Thale's Theorem.
3) To apply the property of the centroid in a triangle
4) To prove that a point represents the centroid of a triangle by applying the converse theoem of centroid.
5) To divide a segment into n equal parts.
6) To locate a point on a segment or a ray that divides the segment / ray in a given ratio.
7) To enlarge or reduce a geometric figure given a center and a scale factor.
8) To write the program of constuction.
9) To find the locus of a variable point with respect to fixed elements.

For the week of Jan. 23-Jan 27, we will cover the first eight objectives.
For the week of Jan.30 -Feb. 6, we will discuss the ninth objective.
 

Welcome!

Welcome to the new year 2011-2012