State and prove the Basic Proportionality Theorem (Thales's Theorem). Also, in triangle ABC, DE is parallel to BC inters

Question

State and prove the Basic Proportionality Theorem (Thales's Theorem). Also, in triangle ABC, DE is parallel to BC intersecting AB at D and AC at E. If AD = 1.5 cm, DB = 3 cm, and AE = 1 cm, find the length of EC.

Accepted Answer

Statement of Thales's Theorem:
If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio. Proof: Given: In $	riangle ABC$, $DE \parallel BC$, intersecting $AB$ at $D$ and $AC$ at $E$. To Prove: $\frac{AD}{DB} = \frac{AE}{EC}$ Construction: Join $B$ to $E$ and $C$ to $D$. Draw $DM \perp AC$ and $EN \perp AB$. Proof Steps: The area of $	riangle ADE$ is given by: $$	ext{Area}(	riangle ADE) = \frac{1}{2} 	imes 	ext{base} 	imes 	ext{height} = \frac{1}{2} 	imes AD 	imes EN$$ The area of $	riangle BDE$ is: $$	ext{Area}(	riangle BDE) = \frac{1}{2} 	imes DB 	imes EN$$ Dividing the two areas: $$\frac{	ext{Area}(	riangle ADE)}{	ext{Area}(	riangle BDE)} = \frac{\frac{1}{2} 	imes AD 	imes EN}{\frac{1}{2} 	imes DB 	imes EN} = \frac{AD}{DB} \quad 	ext{--- (Equation 1)}$$ Similarly, considering $AE$ and $EC$ as bases with height $DM$: $$\frac{	ext{Area}(	riangle ADE)}{	ext{Area}(	riangle DEC)} = \frac{\frac{1}{2} 	imes AE 	imes DM}{\frac{1}{2} 	imes EC 	imes DM} = \frac{AE}{EC} \quad 	ext{--- (Equation 2)}$$ Since $	riangle BDE$ and $	riangle DEC$ lie on the same base $DE$ and between the same parallel lines $DE$ and $BC$, their areas are equal: $$	ext{Area}(	riangle BDE) = 	ext{Area}(	riangle DEC) \quad 	ext{--- (Equation 3)}$$ From Equations (1), (2), and (3), we conclude that: $$\frac{AD}{DB} = \frac{AE}{EC}$$
Hence proved. --- Numerical Solution: Given data: $AD = 1.5	ext{ cm}$ $DB = 3	ext{ cm}$ $AE = 1	ext{ cm}$ $EC = ?$ Formula:
According to Thales's Theorem: $$\frac{AD}{DB} = \frac{AE}{EC}$$ Substitution: $$\frac{1.5}{3} = \frac{1}{EC}$$ Calculation: $$1.5 	imes EC = 3 	imes 1$$ $$1.5 	imes EC = 3$$ $$EC = \frac{3}{1.5}$$ $$EC = 2	ext{ cm}$$ Answer:
The length of $EC$ is $2	ext{ cm}$.

Triangles

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