UPSC CDS 2 2022 Math Paper

Area of similar triangle Theorem:If two triangles are similar, then the ratio of the area of both triangles is proportional to the square of the ratio of their corresponding sides.We have,
⇒DE|BC
⇒ar(△ADE)=ar(△‌ trap.DEBC ‌)
⇒ar(△ADE)=ar(△ABC)−ar(△ADE)
⇒2‌ar(△ADE)=ar(△ABC)‌‌‌ In ‌
‌ In ‌△ADE‌ and ‌△ABC
⇒∠A=∠A‌‌‌‌‌ common] ‌
⇒∠ADE=∠ABC‌‌‌ [corresponding angles ‌]
⇒△ADE∼△ABC‌‌[‌ by AA similarity ‌]
By‌area‌ of similar triangle theorem ‌
⇒‌

ar(△ADE)

ar(△ABC)

=‌

AD2

AB2

⇒‌

ar(△ADE)

2‌ar(△ADE)

=‌

AD2

AB2

‌‌‌ by using (1) ‌
⇒‌

1

2

=‌

AD

AB

‌2
Taking‌ square root ‌
⇒‌

1

√2

=‌

AD

AB

⇒‌

1

√2

=‌

AB−BD

AB

⇒‌

1

√2

=1−‌

BD

AB

⇒‌

BD

AB

=1−‌

1

√2

⇒‌

BD

AB

=‌

√2−1

√2

∴ The ratio of BD∕AB is ‌

√2−1

√2