In triangle ABC ,D and E are the mid points of BC and AC respectively prove that the area of triangle ABC is 4 times than triangle ABE
By the midpoint theorem, DE = (1/2)AB and DE || AB
so we have similar triangles EDC and ABC
let the altitude from C to DE be h
then by similar triangles the altitude from C to AB = 2h
and AB = 2DE
area of triangle DEC = (1/2)(DE)h
area of triangle ABC = (1/2)(AB)(2h)
= (1/2)(2DE)(2h
= 2(DE)h = 4[ (1/2)(DE)h ]
To prove that the area of triangle ABC is four times the area of triangle ABE, we can use the concept of triangle area ratios by considering the midpoints D and E.
Let's denote the area of triangle ABC as [ABC], and the area of triangle ABE as [ABE].
First, we need to establish a relationship between the areas of two triangles with respect to their sides. This can be done by considering the property of similar triangles.
Using the midpoints D and E, we can observe that triangle ABC is divided into four congruent smaller triangles: ABD, DEA, EBC, and CDE. Each of these four triangles is similar to triangle ABC.
Since triangles ABD, DEA, and EBC share a common vertex A, we can conclude that they are also similar to triangle ABE.
Now, let's consider the ratios of the sides of triangles ABC and ABE. We know that the length of AD is half the length of AB since D is the midpoint of BC. Similarly, the length of AE is half the length of AB since E is the midpoint of AC.
Using the fact that triangle ABD is similar to triangle ABC, we can write the proportion:
AB/AD = AC/AE
Substituting the lengths:
AB/(AB/2) = AC/(AC/2)
2 = 2
We see that the proportion is equal, which implies that triangle ABD is similar to triangle ABC.
Now, let's consider the ratio of the areas of triangles ABC and ABD:
[ABC]/[ABD] = (AB/AD)^2
Substituting the lengths:
[ABC]/[ABD] = (AB/(AB/2))^2
[ABC]/[ABD] = (2)^2
[ABC]/[ABD] = 4
Since triangle ABD is similar to triangle ABC, their areas are in the same ratio. Therefore, if [ABC]/[ABD] = 4, we can conclude that:
[ABC] = 4 * [ABD]
However, triangle ABD is a component of triangle ABE:
[ABD] = [ABE]
Substituting this back into our equation:
[ABC] = 4 * [ABE]
Hence, we have proved that the area of triangle ABC is four times the area of triangle ABE.