7. State and prove the basic proportionality theorem.
आधारभूत आनुपातिकता प्रमेय का कथन लिखकर सिद्ध कीजिए।
Answers
Answered by
2
Basic Proportionality Theorem states that, if a line is parallel to a side of a triangle which intersects the other sides into two distinct points,then the line divides those sides of the triangle in proportion.
Let ABC be the triangle.
The line l parallel to BC intersect AB at D and AC at E.
To prove
DB
AD
=
EC
AE
Join BE,CD
Draw EF⊥AB, DG⊥CA
Since EF⊥AB,
EF is the height of triangles ADE and DBE
Area of △ADE=
2
1
× base × height=
2
1
AD×EF
Area of △DBE=
2
1
×DB×EF
areaofΔDBE
areaofΔADE
=
2
1
×DB×EF
2
1
×AD×EF
=
DB
AD
........(1)
Similarly,
areaofΔDCE
areaofΔADE
=
2
1
×EC×DG
2
1
×AE×DG
=
EC
AE
......(2)
But ΔDBE and ΔDCE are the same base DE and between the same parallel straight line BC and DE.
Area of ΔDBE= area of ΔDCE ....(3)
From (1), (2) and (3), we have
DB
AD
=
EC
AE
Hence proved.
Let ABC be the triangle.
The line l parallel to BC intersect AB at D and AC at E.
To prove
DB
AD
=
EC
AE
Join BE,CD
Draw EF⊥AB, DG⊥CA
Since EF⊥AB,
EF is the height of triangles ADE and DBE
Area of △ADE=
2
1
× base × height=
2
1
AD×EF
Area of △DBE=
2
1
×DB×EF
areaofΔDBE
areaofΔADE
=
2
1
×DB×EF
2
1
×AD×EF
=
DB
AD
........(1)
Similarly,
areaofΔDCE
areaofΔADE
=
2
1
×EC×DG
2
1
×AE×DG
=
EC
AE
......(2)
But ΔDBE and ΔDCE are the same base DE and between the same parallel straight line BC and DE.
Area of ΔDBE= area of ΔDCE ....(3)
From (1), (2) and (3), we have
DB
AD
=
EC
AE
Hence proved.
Attachments:
Similar questions