State and prove Basic Proportionality theorem.
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Step-by-Step Solution
Step 1: State the Basic Proportionality Theorem
The Basic Proportionality Theorem, also known as Thales Theorem, describes a fundamental property of triangles when a line parallel to one of its sides intersects the other two sides. It states that the line divides the other two sides proportionally.
Step 2: Setup for Proof
Consider a triangle ABC. Let DE be a line segment drawn parallel to side BC, intersecting sides AB at D and AC at E. We need to prove that the ratio of the segments AD to DB is equal to the ratio of AE to EC.
Step 3: Construction
To prove the theorem, we make two constructions. First, we join B to E and C to D. Second, we draw perpendiculars from E to AB (let's call the foot of the perpendicular M) and from D to AC (let's call the foot of the perpendicular N). These perpendiculars will be used to calculate the areas of the triangles.
Step 4: Calculate Area Ratios
The area of a triangle is given by half times base times height. Using EM as the height for triangles ADE and BDE, we find the ratio of their areas is AD/DB. Similarly, using DN as the height for triangles ADE and CDE, the ratio of their areas is AE/EC.
Step 5: Equate Area Ratios
Triangles BDE and CDE lie between the same parallel lines DE and BC, and they share the same base DE. Therefore, their areas are equal. By substituting this equality into our previous area ratios, we conclude that AD/DB must be equal to AE/EC, thus proving the Basic Proportionality Theorem.