Assignment (Basic Proportionality Theorem and its Converse)

1. In a triangle ABC, D and E are points on the sides AB and AC respectively such that DE BC.
i) If AD=4, AE=8, DB = x-4, and EC=3x-19, find x.
ii) If AD/BD = 4/5 and EC=2.5 cm, find AE.
iii) If AD=x, DB=x-2, AE=x+2 and EC=x-1, find the value of x.
iv) If AD=2.5 cm, BD=3.0 cm and AE=3.75 cm, find the length of AC.

2. In a triangle ABC, D and E are points on the sides AB and AC respectively. For each of the following show that DE BC.
i) AB = 5.6 cm, AD = 1.4 cm, AC = 7.2 cm and AE = 1.8 cm
ii) AD = 5.7 cm, BD = 9.5 cm, AE = 3.3 cm and EC = 5.5 cm

3. In a triangle ABC, P and Q are points on sides AB and AC respectively, such that PQ BC. If AP = 2.4 cm, AQ = 2.0 cm, QC = 3.0 cm and BC = 6.0 cm, Find AB and PQ.

4. In a triangle ABC, D and E are points on AB and AC respectively such that DE BC. If AD = 2.4 cm, AE = 3.2 cm, DE = 2.0 cm and BC = 5.0 cm, find BD and CE.

5. Using basic proportionality theorem, prove that any line parallel to the parallel sides of a trapezium divides the non parallel sides proportionally.

6. In the given figure, PA, QB and RC are each perpendicular to AC.
Prove that 1/x + 1/z = 1/y.

7. ABCD is a trapezium with AB CD. The diagonals AC and BD intersect each other at O. If AO = 2x+4, OC=2x-1, DO=3 and OB = 9x-21, Find x.

8. Prove that the line segments joining the mid points of adjacent sides of a quadrilateral form a parallelogram.

Activity related to Pythagoras Theorem

This project is contributed by Mayank X-B
Aim-To prove Pythagoras theorem by paper cutting and pasting.
Material Required-Thermocol,coloured sheets,cutter,scissors,sketchpens
ruler,fevistick .
Procedure-1 On a coloured sheet of paper draw triangle ABC with AB=3
units, BC=4 units and AC=5 units. Cut it out. (Taking 1 unit=1.5 inches)

2 Paste this triangle on a coloured sheet of paper covering a
thermocol

.
3 On AB paste a square ABDE of side=3 units

.
4 On BC paste a square BCFG of side=4 units.

5 On AC paste a square ACHI of side=5 units.


6 Make replicas of squares ABDE and BCFG.

7 Cut the replica of ABDE into 9 small squares each of area=1 sq unit.

8 Cut the replica of BCFG into 16 small squares each of area=1 sq unit.

9 Paste the unit squares obtained in step-7 and step-8 on square
ACHI.

Observation-The unit squares overlap square ACHI completely.

Result-Therefore, area of square ABDE + area of square BCFG = area of square
ACHI , ie ,
AB^2+ BC^2 = AC^2

Assignment-A.P

1. Do any two consecutive terms of an A.P can be equal ?
2. If the first term of an A.P is a and the common difference is d, what will be the 4thterm of the sequence?
3. How many 2-digit numbers are divisible by 3?
4. If in an A.P, first term is 5 , common difference is 5 and the last term is 60, then find the sum of first n terms.
5. Find the next term of given A.P. √8, √18, √32, √50,………..
6. Find the next term of the given A.P.3,3,3,3,3,3,3,………….
7. Find the first negative term of the A.P. 114, 109, 104,…
8. Which term of A.P. -2, 3, 8, 13,… is 78?
9. Which term of sequence 22, 19, 16,… is the first negative term?
10. Find the 6th term from end of the A.P. 17, 14, 11…-40.
11. Find the sum of first 40 positive integers divisible by 3.
12. How many terms of the A.P. 3,5,7,9,… must be added to get the sum 120?
13. Find the number of terms in an A.P, in which the a=5, common difference=3 and the last term =83
14. The sum of the first 30 terms of an A.P. is 1635.if its last term is 98,find the first term and the common difference of the given A.P.
15. If the 3rd and the 9th terms of an A.P. are 4 and -8 respectively. Which term of the A.P. will be 0?
16. If the mth term of an AP is 1/n and the nth term is 1/m. Show that the sum of (mn) terms is (mn+1)/2
17. If the sum of m terms of an AP is the same as the sum of its n terms. Show that the sum of its (m+n) terms is zero.
18. If the numbers a, b, c, d, e forms an AP , then find the value of a – 4b + 6c – 4d + e
19. The sum of n, 2n, 3n terms of an AP are S1 , S2, S3 respectively. Prove that
S3 = 3 ( S2 - S1)
20. Find the four number in AP whose sum is 20 and sum of whose squares is 120.
21. Find the number of integers between 50 and 500 divisible by 5.
22. Find the number of terms in the sequence
20 , 19 1/3 , 18 2/3 ,......................... of which the sum is 300. Explain the double answer.
23. In an AP, the sum of first n terms is (3n2 + 5n ) / 2. Find the 25th term.
24. The sum of n terms of three AP are S1 , S2 and S3. The first term of each is unity and the common difference are 1,2 and 3 respectively. Prove that S1 + S3 = 2S2

Assignment-Coordinate Geometry

1.If the vertices of a triangle are (2,4), (5,k) and (3,10) and its area is
15 sq cm. Find the value of k.
2. Find the distance of a point (2,3) from origin.
3. The three vertices of a rhombus, taken in order are (2,-1), (3,4) and
(-2,3). Find the fourth vertex.
4. Are (4,8) ,(7,5) and (1,-1) the vertices of a right triangle?
5. There are two points A(-2,-3) and B(2,3). Is the distance of these
points from origin is same?
6. If (-2,-1), (a,0), (4,3) and (1,2) are the vertices of a parallelogram.
Find the value of a.
7. Find the distance between the points (-4,p) and (5,q).
8. Find the area of the triangle formed by the points P(2,1), Q(6,1) and
R(2,4).
9. On which axis the point (-5,0) lie?
10. Find the value of p for which the points (-1,3) , (2,p) and (5,-1) are
collinear.
11. Find the relation between x and y if the points (x,y),(1,2) and (7,0) are
collinear.
12. Find the coordinates of the point which divides the line segment
joining the points (4, – 3) and (8, 5) in the ratio 3 : 1 internally.
13. If A(–5, 7), B(– 4, –5), C(–1, –6) and D(4, 5) are the vertices of a
quadrilateral, find the area of the quadrilateral ABCD.
14. Find a relation between x and y such that the point (x , y) is
equidistant from the points (7, 1) and (3, 5).
15. Find the coordinates of the points of trisection of the line segment
joining the points A(2, – 2) and B(– 7, 4).
16. If A(5, -1), B(-3, -2) and C(-1, 8) are the vertices of triangle ABC, find
the length of median through A and the coordinates of the centroid.
17. Prove that the points (-3, 0), (1, -3) and (4,1) are the vertices of an
isosceles right angled triangle. Find the area of this triangle.
18. Find the area of the triangle formed by joining the mid – points of the
sides of the triangle whose vertices are (0, -1),(2, 1) and (0, 3).
Find the ratio of the area of the triangle formed to the area of the given
triangle.
19. If A( -2, -1) , B( a, 0) C( 4, b) and D(1, 2) are the vertices of a
parallelogram , find the values of a and b.
20. Let the opposite points of a square be ( 3, 4) and (1, -1) . Find the
coordinates of the remaining angular points.
21. If two vertices of an equilateral triangle be ( 0 , 0) , ( 3 , √3 ). Find the
third vertex.
22. If the point C ( -1 ,2) divides internally the line segment joining A(2,5)
and B in the ratio 3:4. Find the coordinates of B.
23. Find the coordinates of the circumcentre of the triangle whose
vertices are (8,6) (8,-2) and (2,-2) .
24. If the vertex of a triangle be (1,1) and the middle points of the sides
through it be ( -2 , 3) and (5,2). Find the other vertices.
25. In the figure BOA is a right triangle and C is the midpoint of hypt
AB.Show that it is equidistant from the vertices O,A and B.

SLIDES ON ARITHMETIC PROGRESSION

PASCAL TRIANGLE

Pascal's triangle is a geometric arrangement of the binomial cofficients in a triangle.The triangle is bordered by ones on the right and left sides, and each interior entry is the sum of the two entries above.

Construction:-
At the top of Pascal's Triangle it have number 1 which is the zeroth row. Add the number directly above and to the left with the number directly above and to the right to find the new value. If either the number to the right or left is not present, substitute a zero in its place.
First Row 0+1=1,1+0=1
Second Row 0+1=1,1+1=2,1+0=1
Third Row 0+1=1,1+2=3,2+1=3,1+0=1
And so on the rows go infinitely.
Some properties:-
Sums of the numbers in Row
The sum of the numbers in any row is equal to nth power of 2, when n is the number of the row.
For example:

and so on
Prime Numbers:-

The 0th element of every row is 1 and if the first element in any row is prime, then all the numbers in that row excluding 1’s are divisible by it .
For example:
In row5( 1 5 10 10 5 1)5,10 are divisible by 5.

Hockey Stick Pattern:-
If a diagonal of numbers of any length is selected starting at any of the 1's bordering the sides of the triangle and ending on any number inside the triangle on that diagonal, the sum of the numbers inside the selection is equal to the number below the end of the selection that is not on the same diagonal itself.

For example:

1+6+21+56 = 84

1+7+28+84+210+462+924 = 1716

1+12 = 13
Powers of 11’s:-