Answer:
Basic proportionality theorem was proposed by a famous Greek mathematician, Thales, hence, it is also referred to as the Thales theorem. According to the famous mathematician, for any two equiangular triangles, the ratio of any two corresponding sides of the given triangles is always the same. Based on this concept, the basic proportionality theorem(BPT) was proposed. It gives the relationship between the sides of any two equiangular triangles.
The concept of Thales theorem has been introduced in similar triangles. If the given two triangles are similar to each other then,
Corresponding angles of both the triangles are equal
Corresponding sides of both the triangles are in proportion to each other
The theorem thus also helps us better understand the concept of similar triangles. Now let us try and understand the Basic Proportionality Theorem.
Statement: The line drawn parallel to one side of a triangle and cutting the other two sides divides the other two sides in equal proportion.
Given: Consider a triangle ΔABC, as shown in the given figure. In this triangle, we draw a line DE parallel to the side BC of ΔABC and intersecting the sides AB and AC at D and E, respectively.
Construction: In the above diagram, create imaginary lines where you can Join C to D and B to E. Draw perpendicular DP perpendicular to AE and EQ perpendicular to AD.
Proof:
Consider the triangles ADE and BDE. Both these triangles are on the same base AB and have equal height EQ.
(Area of ADE)/(Area of BDE) = (1/2 × AD × EQ)/(1/2 × BD × EQ)
(Area of ADE)/(Area of BDE) = AD/BD
Now consider triangles CDE and ADE. Both these triangles are on the same base AC and have equal height DP.
(Area of ADE)/(Area of CDE) = (1/2 × AE × DP)/(1/2 × CE × DP)
(Area of ADE)/(Area of CDE) = AE/CE
Both the triangles BDE and CDE are between the same set of parallel lines.
Area of triangle BDE = Area of triangle CDE
Applying this we have, (Area of triangle ADE)/(Area of triangle BDE) = (Area of triangle ADE)/(Area of triangle CDE)
AD/BD = AE/CE