Question

In the adjoining figure; BD = DC and AE = ED. Prove that area of ΔACE = `1/4` × area of ΔАВС.

Answer 1

Given:

BD = DC, so D is midpoint of BC.

AE = E,D so E is midpoint of AD.

To Prove: Area (ΔACE) = `1/4` × Area (ΔABC).

Proof (Step-wise):

1. Because BD = DC, D is the midpoint of BC.

Hence, triangles ABD and ADC have the same base length on BC (BD = DC) and share the same altitude from A.

So, they have equal areas. 

Therefore, area (ΔADC) = `1/2` × area(ΔABC). 

2. Consider triangle ADC.

Points A, E, D are collinear with E the midpoint of AD (AE = ED). 

Triangles ACE and A D C have the same vertex C and the same altitude from C to the line AD.

So, their areas are proportional to their bases on AD.

Thus, `("Area"(ΔACE))/("Area"(ΔADC)) = (AE)/(AD)`.

Area of a triangle = `1/2` × base × height.

So, area ratios with common height equal base ratios.

3. Since E is the midpoint of AD,

`(AE)/(AD) = 1/2` 

From step 2:

area (ΔACE) = `1/2` × area (ΔADC)

4. Combine steps 1 and 3:

Area (ΔACE) = `1/2` × Area (ΔADC) 

= `1/2` × `1/2` × area (ΔABC) 

= `1/4` × area (ΔABC)