Question

While preparing a Power Point presentation, ΔABC is enlarged along the side BC to ΔAB’C’, as shown in the diagram, such that BC : B’C’ is 3 : 5. Find:

1. AB : BB’
2. length AB, if BB’ = 4 cm.
3. Is ΔABC ~ ΔАВ’C’? Justify your answer.
4. ar (ΔABC) : ar (quad. BB’C’C).

Answer 1

Given: ΔABC is enlarged to ΔAB’C’ with A common and BC : B’C’ = 3 : 5, so the linear scale factor `k = (B’C’)/(BC) = 5/3`.

Step-wise calculation:

1. Relation between AB and BB’:

Because the centre of enlargement is A.

AB’ = k × AB 

= `(5/3) AB`

Hence, BB’ = AB’ – AB

= `(5/3 - 1) AB` 

= `(2/3) AB`

Therefore, AB : BB’

= `AB : (2/3 AB)` 

= 3 : 2

2. Length AB when BB’ = 4 cm:

`BB’ = (2/3)AB = 4`

⇒ `AB = (3/2) xx 4 = 6  cm`

3. Similarity of ΔABC and ΔAB’C’:

Corresponding sides are in the same ratio `k = 5/3 ((AB’)/(AB) = (AC’)/(AC) = (B’C’)/(BC))` and angle A is common.

Thus, ΔABC ∼ ΔAB’C’ triangles are similar by enlargement about A.

4. Area ratio ar(ΔABC) : ar(quad. BB’C’C):

Areas scale by k², so ar(ΔAB’C’) = k² × ar(ΔABC)

= `(5/3)^2 xx ar(ΔABC)`

= `25/9 xx ar(ΔABC)`

Quadrilateral BB’C’C = ar(ΔAB’C’) – ar(ΔABC)

= `(25/9 - 1) ar(ΔABC)`

= `16/9 xx ar(ΔABC)`

Therefore, ar(ΔABC) : ar(BB’C’C)

= `1 : 16/9`

= 9 : 16