Triangle ∆abc has area of s m2. points n and p are taken on rays bc and ca respectively so that the length bn = 2bc and cp = 3ca. find the area of ∆pbn. (hints: find the area of ∆pbc in terms of s by looking for triangles with equal heights. then compare the area of ∆pbc to the area of ∆pnc.)

Question

Triangle ∆abc has area of s m2. points n and p are taken on rays bc and ca respectively so that the length bn = 2bc and cp = 3ca. find the area of ∆pbn. (hints: find the area of ∆pbc in terms of s by looking for triangles with equal heights. then compare the area of ∆pbc to the area of ∆pnc.)

Answer 1

Triangle PBC has the same height as triangle ABC, but the base, PC, is 3 times longer, so the area of the triangle PBC is 3 times the area of the triangle ABC. Since triangle ABC has area S, triangle PBC has area 3S.

Triangle PCN has the same base and height as triangle PBC, so it has the same area. The area of triangle PCN is 3S.
Triangle PBN is made up of triangles PBC and PCN, so the area of triangle PBN is 3S + 3S = 6S m^2.

Answer 2

ABC and PBC have the same height.

PBC has base 3 times ABC, so
area PBC = 3 * area ABC = 3s
See what you can do with that.

Answer 3

To find the area of ∆PBN, we can follow these steps:

Step 1: Find the area of ∆PBC in terms of s.
By looking for triangles with equal heights, we can find the area of ∆PBC.

Let's denote the height of ∆ABC as h.
Since BN is twice the length of BC, we can say that the height of ∆PBN is also twice the height of ∆ABC, which is 2h.
Similarly, since CP is three times the length of CA, the height of ∆PNC is three times the height of ∆ABC, which is 3h.

Now, we can find the area of ∆PBC:

Area of ∆PBC = (base BC) * (height 2h) / 2 = BC * 2h / 2 = BC * h = s/2

So, the area of ∆PBC in terms of s is s/2.

Step 2: Compare the area of ∆PBC to the area of ∆PNC.
Both ∆PBC and ∆PNC share a common base, which is BC. The height of ∆PNC (3h) is three times the height of ∆PBC (2h).

We know that the ratio of the areas of two triangles with the same base is equal to the ratio of their corresponding heights.

Therefore, the ratio of the area of ∆PNC to the area of ∆PBC can be given as: (Area of ∆PNC) / (Area of ∆PBC) = (3h) / (2h) = 3/2.

Step 3: Find the area of ∆PBN.
Since ∆PBN shares the same base (BN) as ∆PBC, but has twice the height (2h), we can say that the area of ∆PBN is twice the area of ∆PBC.

Therefore, the area of ∆PBN = 2 * (Area of ∆PBC) = 2 * (s/2) = s.

So, the area of ∆PBN is equal to s (in terms of s).

In conclusion, the area of ∆PBN is s square units.