BPT -- Basic Proportionality Theorem

Proportionality Theorem states that, A line parallel to one side of a triangle divides the other two sides into parts of equal proportion.

In triangle ABC, a line drawn parallel to BC cuts AB and AC at P and Q respectively.




Let the point P divide AB in the ratio of l: m where l and m are natural numbers. Divide AP into 'l' and PB into 'm' equal parts. Through each of these points on AB, draw lines parallel to BC to cut AC.


Basic Proportionality Theorem (B.P.T.) will be more useful in the topic 'SIMILARITY'.

Division of a line segment into equal parts.
Divide a line segment of length 8.4 cm into 5 equal parts.

AB = 8.4 cm



SECOND METHOD
1. Draw AB = 8.4 cm and through A draw another line AX at an acute angle to AB.
2. With a suitable radius, cut off equal lengths AP, PQ, QR, RS and ST.
3. Join TB. Draw SF, RE, QD and PC parallel to TB to cut AB at F, E, D and C. The line segment AB is divided into five equal parts.
AC = CD = DE = EF = FB
Divide AB = 8.4 cm internally in the ratio of 3 : 2.


1. Draw AB = 8.4 cm and through A draw another line AX at an acute angle to AB.
2. Make ÐABY = ÐBAX so that BY is on the opposite side of AB to that of AX.
3. With suitable radius, cut off equal lengths AH, HJ, JK, KL and LM on AX.
Similarly, with the same radius cut off BP = PQ = QR = RS = ST on BY.
4. Join AT, HS, JR, KQ, LP and MB to cut AB at points C, D, E and F respectively.
AB is divided at E in the ratio of 3 : 2.

• Midpoint Theorem

The straight line joining the mid-points of two sides of a triangle is parallel to and equal to half the third side.

• Converse of mid point theorem - The straight line drawn through the mid point of one side of a triangle, parallel to another, bisects the third side.
• The Intercepts Theorem

If three or more parallel lines make equal intercepts on a transversal, then they make equal intercepts on any other transversal.

• A line parallel to one side of a triangle, divides the other two sides proportionally (Basic Proportionality Theorem).
• Using the above results, we can divide a line segment into a number of equal parts.

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Matrices Worksheets

Matrix Worksheets

1Given the matrices:

Calculate:

(A + B)²;       (A - B)²;       (B)³;        A · B^t · C.

2Prove that: A² − A − 2 I = 0, where:

3 Given the matrix A =  . Find Aⁿ , for n
4Determine the matrix that is needed to premultiply by the matrix
in order to obtain the matrix .
5Find all matrices that commute with the matrix:

6 Determine the matrices A and B that verify the system:

7  A factory produces two models of washing machines, A and B, in three available finishes: N, L and S. Model A is produced in 400 units in Finish N, 200 units in Finish L and 50 units in Finish S. Model B is produced in 300 units in Finish N, 100 units in Finish L and 30 units in Finish S. Finish N takes 25 hours of workshop time to complete and 1 hour of administration. Finish L takes 30 hours of workshop time and 1.2 hours of administration. Finally, Finish S takes 33 hours of workshop time and 1.3 hours of administration.
1.Represent the information in two matrices.
2.Find a matrix that expresses the hours of workshop and administration time needed for each of the models.
8A furniture company manufactures three different models of shelves: A, B and C. Each of these models is offered in both large and small sizes. Everyday, the factory produces 1,000 large and 8,000 small shelves in Type A, 8,000 large and 6,000 small shelves in Type B and 4,000 large and 6,000 small shelves in Type C. Regardless of the type, each large shelf has 16 screws and 6 supporting brackets while each small shelf has 12 screws and 4 supporting brackets.

1Represent this information in two matrices.

2Find a matrix that represents the quantity of screws and supports brackets necessary for daily production of each of the six model-sized shelves.

9Given the matrices:

Calculate the value of X in the following equations:

Prove that: A² − A − 2 I = 0, where:

Given the matrix A = . Find Aⁿ , for n

Can you answer these questions

1)  IF 3 CONSEQUTIVE BINOMIAL COEFFICIENTS : ⁿC_r-1 , ⁿC_r , ⁿC_r+1 : ARE IN A.P., THEN

  

                                                   r  =  1/2 (n    (n + 2) )

 

    BE AWARE THAT : 3 CONSEQUTIVE BINOMIAL COEFFICIENTS ARE NEVER IN G.P. OR H.P. AND THAT 4 CONSEQUTIVE BINOMIAL COEFFICIENTS ARE NEVER IN A.P.

 

 

2)  THE TERM IN x^m IN THE EXPANSION OF ( ax^p + b/x^q)ⁿ IS T_r+1 WHERE :

 

                                                   r  = (np - m) / (p + q)

 

     OBVIOUSLY, THE TERM INDEPENDENT OF 'x' IS T_r+1 WHERE :

 

                                                   r  =  np / (p +q)

 

 

3)  THE NO. OF DISTINCT TERMS IN THE EXPANSION OF :

      (a₁ + a₂ + a₃ + ................ + a_m)ⁿ IS :

 

                                          ^n+m-1 C _m-1  

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