Maths question (triangle)

hope you can learn more than copy, friends!

if "a" = shorter side, "b" = longer side, "A" = area of the triangle
then A = (a x b) / 2, i.e. ab/2

a & b differ by 6, i.e. b-a=6 => b=6+a

if b+3, a-2, then the area is A-5,
i.e. ((b+3)x(a-2))/2=A-5
=> ((b+3)x(a-2)=(A-5)x2
=> (b+3)x(a-2)=2A-10
=> (b+3)x(a-2)=2(ab/2)-10
=> (b+3)x(a-2)=ab-10
=> ab+3a-2b-6=ab-10
=> 3a-2b=ab-10-ab+6
=> 3a-2b= -4
=> 3a-2(6+a)= -4
=> 3a-12-2a= -4
=> a=8

=> b=6+a =6+8 =14

therefore, A=ab/2 =8x14/2 =56

56 is the answer, right?

Let the shorter length of the triangle be x,
>> the longer length is (6+x)
(Since two perpendicular sides of a right-angled triangle differ by 6)

>> Area= x(6+x)/2

If the length of the longer of these two sides is increased by 3 centimetres
>> (6+x)+3

The length of the shorter of these two sides is decreased by 2 centimetres
>> x-2

Then the area of the right-angled triangle formed is decreased by 5 centimetres
>> Area= x(6+x)/2 - 5

Therefore, x(6+x)/2 - 5 = ((6+x)+3) (x-2)/2
(6x+x^2)/2 - 5 = (9+x)(x-2)/2
6x+x^2-10=9x+x^2-18-2x
6x-10=7x-18
>> x= 8 cm

What is the dimension of the original triangle?
>> Because 8^2+14^2=260
>> Dimension: 8*14*sqr root(260)

2007-05-02 21:48:10 補充：
註: dimension 有兩個解釋，不過在數學上通當會用「尺寸」，即長，寬，厚，高等等。所以我認為問題的答案應該是 8*14*sqr root(260) （即是底，高，斜邊）第二個解釋是 area。如果真的如此，答案才是 56 cm2

Let the length of the shorter side be X cm.
Then the length of the longer side would be (X+6) cm.

Thus, area of the triangle = (0.5)(X)(X+6) = 0.5X^2 + 3X ----- (1)

Area of the other triangle = (0.5)(X+6+3)(X-2) = (0.5)(X+9)(X-2)
=0.5(X^2+9X-2X+18)=0.5(X^2+7X-18)
= 0.5X^2 +3.5X -9 ---- (2).

From given information, we have area of the other triangle - area of the other triangle = 5
From (1) - (2), we have [0.5X^2 + 3X] - [0.5X^2 +3.5X -9 ] = 5
-0.5X + 9 = 5
-0.5X = 5 - 9 = -4
X = 8.

The shorter side of the original triangle is 8 cm and therefore the length of the prependicular one is 14 cm. Then the length of the longest one of the triangle would be Square Root of (8x8 + 14 x14) = 2sqr(65).

2007-05-01 15:55:34 補充：
Area of the original triangle = (0.5)(8)(14) = 56 cm^2.