Ex 6.5,3
In figure, ABD is a triangle right angled at A and AC β₯ BD. Show that
AB2 = BC . BD
Given:
ABD is a triangle right angled at A .
& AC β₯ π΅π·
To prove: AB2 = BC . BD
i.e. π΄π΅/π΅π· = π΅πΆ/π΄π΅
Proof:
From theorem 6.7,
If a perpendicular is drawn from the vertex of the right angle to
the hypotenuse then triangles on both sides of the
Perpendicular are similar to the whole triangle and to each other
So, Ξ BAD βΌ Ξ ACB
If two triangles are similar ,
then the ratio of their corresponding sides are equal
π΅π΄/π΅πΆ=π΅π·/π΅π΄
BA Γ BA = BD Γ BC
BA2 = BD Γ BC
i.e. AB2 = BD Γ BC
Hence proved
Ex 6.5,3
In figure, ABD is a triangle right angled at A and AC β₯BD. Show that
(ii) AC2 = BC . DC
We need to prove: AC2 = BC . DC
i.e. π΄πΆ/π·πΆ = π΅πΆ/π΄πΆ
From theorem 6.7,
If a perpendicular is drawn from the vertex of the right angle to
the hypotenuse then triangles on both sides of the
Perpendicular are similar to the whole triangle and to each other
So, Ξ BCA βΌ Ξ ACD
If two triangles are similar ,
then the ratio of their corresponding sides are equal
π΅πΆ/π΄πΆ=πΆπ΄/πΆπ·
BC Γ CD = AC Γ CA
BC Γ CD = AC2
AC2 = BC Γ CD
Hence proved
Ex 6.5,3
In figure, ABD is a triangle right angled at A and AC β₯BD. Show that
(iii) AD2 = BD . CD
We need to prove: AD2 = BD . CD
i.e. π΄π·/πΆπ· = π΅π·/π΄π·
From theorem 6.7,
If a perpendicular is drawn from the vertex of the right angle to
the hypotenuse then triangles on both sides of the
Perpendicular are similar to the whole triangle and to each other
So, Ξ DAB βΌ Ξ DCA
If two triangles are similar ,
then the ratio of their corresponding sides are equal
π·π΄/π·πΆ=π·π΅/π·π΄
DA Γ DA = DB Γ DC
DA2 = DB Γ DC
AD2 = BD Γ CD
Hence proved

Made by

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths and Science at Teachoo.