3U question (1 Viewer)

[apollo1]

A series of lines are drawn on a Cartesian diagram such that y = nx, where n can be one of 0,1,2,3,4,5,6,7,8,9.
Triangles are formed using two of the above lines and the line x = 1.

(a) If n < 4, how many such triangles are formed?
(b) How many triangles formed for n < 10?
(c)
(i) How many different pairs of triangles can be formed for n < 10?
(ii) How many of these pairs of triangles have no common area?

this is from my school's 2007 3U trial. culd sumone help

 

[largarithmic]

A series of lines are drawn on a Cartesian diagram such that y = nx, where n can be one of 0,1,2,3,4,5,6,7,8,9.
Triangles are formed using two of the above lines and the line x = 1.

(a) If n < 4, how many such triangles are formed?
(b) How many triangles formed for n < 10?
(c)
(i) How many different pairs of triangles can be formed for n < 10?
(ii) How many of these pairs of triangles have no common area?

this is from my school's 2007 3U trial. culd sumone help

(a) Each triangle is formed by choosing 2 of those lines and the line x = 1. So if n < 4, n is one of 0,1,2,3 so there are 4C2 = 6 triangles
(b) Same as (a) except n can be one of 10 things. So 10C2 = 45 triangles can be formed.

(c)(i) We choose 2 of those triangles above, so 45C2 = 990.

(ii) This question is kinda ambiguous because its unclear what "no common area" means. If "no common area" means "common area = 0", then you have to allow pairs of triangles that share a common side; if "no common area" means "no points in common" then you cant have a side in common. Given though that all possible triangles have a common point anyway (the origin) I interpret it as the first one.

Either the triangles have one common side or no common sides; note that the common side can't be along the line x = 1 or they'd have a common area. (Just think about it visually). If they have one common side, you need 3 lines out of "y=nx"; the middle value of n you chose is the common side, and the bigger/smaller values are the noncommon nonx=1 sides of the two triangles. This gives 10C3 = 120 pairs.

If the triangles have no common sides, then you actually are choosing 4 lines out of "y=nx", and these 4 lines uniquely determine a pair of triangles with no common side or area. (essentially, the top two lines you chose form one triangle, the bottom two form the other). This means there are 10C4 = 210 ways.

Adding these together the answer should be 330.