Theorems for TRIANGLES...

THEOREM:1(FUNDAMENTAL THEOREM OF PROPORTIONALITY)

STATEMENT: A line drawn parallel to a side of a triangle intersects the other two sides in two distinct points so that each of these sides becomes union of two line segments.The lengths of line segments of each side lying in the same semi-plane of that line are proprtional to the lengths of the corresponding sides.

DATA: In Triangle ABC, l is parallel to line segment BC. l intersects line segment AB and line segment AC in D and E respectively.

TO PROVE: (1) AD/AB=AE/AC
(2)BD/AB=EC/AC

PROOF:Construct Line Segment BE and Line Segment CD . Also Let Line segment EF be Perpendicular to Line AB and Line Segment DN be Perpendicular to Line AC. F belongs to Line AB and N belongs to Line AC.

Area of Triangle ADE=1/2 * base * Altitude.

=1/2 * AD * EF ..........................................................................eq.1

ARea of Triangle DBE = 1/2 * base * Altitude.

=1/2 * BD * EF ..........................................................................eq.2




More theorems...7,8,9,10,11.(COMING SOON!!!!)

CHAPTER:1-IMP SUMS

Q. If A gives 1/6th part of the amount with him to B, the Amounts with both becomes equal. If B gives Rs.16 to A, then A's Amount becomes two and a half that with B. Find the amount with each.

ANS. Let the amount with A be Rs.x and the amount with B be Rs.y.

According to the First Condition,
x-x/6 = y+x/6
Therefore, (6x-x)/6 = (6y+x)/6
Therefore, 6x-x = 6y+x
Therefore, 6x-x-6y-x = 0
Therefore, 6x-6y-2x = 0
Therefore, 4x-6y = 0 ...........................eq.1

According to the Second Condition,
x+16 = 5/2 (y-16)
Therefore, 2x+32 = 5y-80
Therefore, 2x-5y = -80-32
Therefore, 2x-5y = -112......................eq.2

Multiplying eq.1 by 2, and then eliminating(subtarction),

We Get...4y = 224
Therefore, y = 224/4
Therefore, y = 56

Putting y=56 in eq.1, we get...

4x-6y=0
Therefore, 4x-6(56)=0
Therefore, 4x-336=0
Therefore, 4x=336
Therefore, x=336/4
Therefore, x=84

ANS: The Amount with A is Rs.84 and The Amount with B is Rs.56