Question

In adjoining figure, PQ ⊥ BC, AD ⊥ BC then find following ratios.

1. `("A"(∆"PQB"))/("A"(∆"PBC"))`
2. `("A"(∆"PBC"))/("A"(∆"ABC"))`
3. `("A"(∆"ABC"))/("A"(∆"ADC"))`
4. `("A"(∆"ADC"))/("A"(∆"PQC"))`

Answer 1

(i) In ∆PQB and ∆PBC,

ΔPQB and ΔPBC have same height PQ.   ...(Given)

Areas of triangles with equal heights are proportional to their corresponding bases.

∴ `("A"(∆"PQB"))/("A"(∆"PBC")) = "BQ"/"BC"`

(ii) In ∆PBC and ∆ABC,

∆PBC and ∆ABC have same base BC.   ...(Given)

Areas of triangles equal bases are proportional to their corresponding heights.

∴ `("A"(∆"PBC"))/("A"(∆"ABC")) = "PQ"/"AD"`

(iii) In ∆ABC and ∆ADC,

∆ABC and ∆ADC have same height AD.   ...(Given)

Areas of triangles with equal heights are proportional to their corresponding bases.

∴ `("A"(∆"ABC"))/("A"(∆"ADC")) = "BC"/"DC"`

(iv) In ∆ADC and ∆PQC,

Ratio of areas of two triangles is equal to the ratio of the products of their bases and corresponding heights.

∴ `("A"(∆"ADC"))/("A"(∆"PQC")) = "DC × AD"/"QC × PQ"`