Question

if ab:cd=(a+b)²:(c+d)², then show that a:b=c:d or a:b=d:c.
please solve this Asap​

Answer 1

Step-by-step explanation:

Given :-

ab:cd=(a+b)²:(c+d)²

To find :-

Show that a:b=c:d or a:b=d:c.

Solution :-

Given that :-

ab:cd=(a+b)²:(c+d)²

Extremes = ab and (c+d)²

Product of extremes = ab×(c+d)²

Means = cd and (a+b)²

Product of means = cd×(a+b)²

Since they are in proportion

Product of extremes= Product of means

=> ab×(c+d)² = cd×(a+b)²

=> ab ( c²+2cd+d² ) = cd (a²+2ab+b²)

=> abc²+2abcd+abd²=a²cd+2abcd+b²cd

=>abc²+2abcd+abd²-a²cd-2abcd-b²cd = 0

=> abc² + abd² - a²cd - b²cd = 0

=> (abc² - a²cd ) + ( abd² - b²cd ) = 0

=> ac ( bc - ad ) + bd ( ad - bc ) = 0

=> ac ( bc - ad ) - bd ( bc - ad ) = 0

=> ( ac - bd ) ( bc - ad ) = 0

=> ac - bd = 0 ( or ) bc - ad = 0

=> ac = bd (or) bc = ad

=> a/b = d/c (or) b/a = d/c

=> a/b = d/c (or) a/b = c/d

=> a : b = d : c (or) a : b = c : d

Hence , Proved.

Answer :-

If ab:cd=(a+b)²:(c+d)², then a:b=c:d (or) a:b=d:c.

Used formulae:-

• Equality of ratios is Proportion.

• In Proportion The product of means is equal to the product of extremes.

• If a,b,c,d are in proportion then a:b = c:d => bc = ad .

• The symbol :: is used for Proportion and it is read as ' is as '.