state and prove basic proportionally theorem
Answers
Answer:
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.
Step-by-step explanation:
Answer:
Step-by-step explanation:
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 AD / DB = AE / EC
Join BE,CD
Draw EF⊥AB, DG⊥CA
Since EF⊥AB,
EF is the height of triangles ADE and DBE
Area of △ADE= 1/2 × base × height
= 1/2 AD×EF
Area of △DBE= 1/2 ×DB×EF
areaofΔADE / areaofΔDBE
= 1/2 × AD × EF / 1/2 × DB × EF
= AD/DB ........(1)
Similarly,
areaofΔADE / areaofΔDCE
= 1/2 × AE × DG / 1/2 × EC × DG
= AE/EC ......(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
AD /DB = AE / EC
Hence proved.