Question

The ratio between the areas of two similar triangles is 16 : 25. State the ratio between their :

1. perimeters.
2. corresponding altitudes.
3. corresponding medians.

Answer 1

ΔABC ∼ ΔDEF, AL ⊥ BC and DM ⊥ EF and AP and DQ are the medians and also area ΔABC : area ΔDEF = 16 : 25

∵ ΔABC ~ ΔDEF   ...(Given)

∴ `(Area  ΔABC)/(Area  ΔDEF) = (AB^2)/(DE^2)`

`\implies (AB^2)/(DE^2) = 16/25 = (4)^2/(5)^2`

∴ `(AB)/(DE) = 4/5` or AB : DE = 4 : 5  ...(i)

∵ ΔABC ∼ ΔDEF

∴ ∠B = ∠E and `(AB)/(DE) = (BC)/(EF)`  ...(i)

i. ∵ ΔABC ∼ ΔDEF

∴ `(AB)/(DE) = (BC)/(EF) = (CA)/(FD)`

= `(AB + BC + CA)/(DE + EF + FD)`

= `4/5`  ...[From (i)]

∴ The ratio between two perimeters = 4 : 5

ii. Now, in ΔABC and ΔDEM,

∴ ∠B = ∠E, ∠L = ∠M  ...(Each 90°)

∴ ΔABL ∼ ΔDEM   ...(AA criterion of similarity)

∴ `(AB)/(DE) = (AL)/(DM) = 4/5`  ...[From (i)]

∵ AL : DM = 4 : 5

iii. ΔABC ∼ ΔDEF, ∠B = ∠E and

`(AB)/(DE) = (BC)/(EF) = (2BP)/(2EQ) = (BP)/(EQ)`

∴ ΔABD ∼ ΔDEQ

∴ `(AB)/(DE) = (AP)/(DQ) = 4/5`  ...[From (i)]

∴ AB : DE = 4 : 5