Question

(a) Prove that if a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then the other two sides are divided in the same ratio.

(b) As shown in the given figure, a girl of height 90 cm is walking away from the base of a lamp post at a speed of $1 \cdot 2$ m/s. If the lamp is $3 \cdot 6$ m above the ground, find the length of her shadow after 4 seconds.

Approach

Prove that if a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then the other two sides are divided in the same ratio. This is the Basic Proportionality Theorem (Thales' theorem). The proof involves drawing a correct figure, stating given and to prove, constructing perpendiculars from the parallel line to the base, and using area ratios to establish the proportionality.

Step-by-step working

1
Draw correct figure with triangle $A BC$ , line $D E ∥ BC$ intersecting $A B$ at $D$ and $A C$ at $E$
1 mark
2
State Given: $D E ∥ BC$ . To Prove: $\frac{A D}{D B} = \frac{A E}{EC}$ . Construction: Draw perpendiculars from $D$ and $E$ to $BC$
1 mark
3
Use area of triangles: $\frac{[ A D E ]}{[ B D E ]} = \frac{A D}{D B}$ and $\frac{[ A D E ]}{[ C D E ]} = \frac{A E}{EC}$ . Since $[B D E] = [C D E]$ (equal heights from $D E$ ), we get $\frac{A D}{D B} = \frac{A E}{EC}$
3 marks

Concepts used

trianglesbasic proportionality theoremsimilar trianglesproveshadow problem

Approach

Step-by-step working

Concepts used

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