Question

show that area of the triangle with the vertices [(a+1)(a+2),a+2],[(a+2),(a+3),a+3] and [(a+3)(a+4),a+4]is independent of a.

Answer 1

help pls

Answer 2

Are these like x,y,z coordinates?

Answer 3

yes

Answer 4

Hmm, I'm not really sure how to do this, but heres an idea. Use the 3D distance formula to find the length between the 3 points, than use heron's formula to find the area. It might be really messy, so there is probably an easier way.\[d = \sqrt {\Delta^2 x + \Delta^2 y + \Delta^2 z} = \sqrt {(x_2-x_1)^2 + ... }\]\[A = \frac {1}{4} \sqrt { (a^2 + b^2 + c^2)^2 - 2(a^4 + b^4 + c^4)}\]

Answer 5

but rogue there are only 2 points read carefully

Answer 6

Oh, so its just x,y coordinates?

Answer 7

If so, then you can still use the same method I said, only theres no delta^z in the distance formula.

Answer 8

phi wats ur opinion?

Answer 9

dumbcow can u hlp?

Answer 10

so this is in xy plane?
yeah use distance formula to get side lengths

Answer 11

Then to get height, take one side and find line perpendicular to it that goes through opposite vertex

Answer 12

wat abt taking determinants?

Answer 13

yeah there prob is a way to get area using matrices...i don't remember how though

Answer 14

show that area of parallelogram formed by lines 3y-2x=a ,2y-3x+a=0,2x-3y+3a=0 and 3x-2y-2a=0 is (2a^2/5)

Answer 15

herons formula should work ok

Answer 16

Area = base*height
find where lines intersect, use those points to get side lengths

Answer 17

a line L intersects the three sides BC,CA and AB of a triangle ABC at P,Q and R .then show that (BP/BC).(CQ/QA).(AR/RB)=-1

Answer 18

how can a line intetsect all 3 sides of a triangle ??

Answer 19

DONT KNOW

Answer 20

You can use the area of a triangle defined by 3 points (x1,y1), (x2,y2), (x3,y3)
is given by
\[A=\frac{1}{2}\left| \left[\begin{matrix}x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{matrix}\right] \right|\]

btw, the vertices [(a+1)(a+2),a+2],[(a+2),(a+3),a+3] and [(a+3)(a+4),a+4]
has a typo, in that the second point is [[(a+2)(a+3),a+3] (only 2 points, obviously)

Answer 21

Here is how I would do it