Question

In a δabc, p and q are points on sides ab and ac respectively, such that pq || bc. if ap = 2.4 cm, aq = 2 cm, qc = 3 cm and bc = 6 cm, find ab and pq.

Answer 1

The value of AB is 2.8and the value of PQ will be 3cm

Answer 2

$\Large{\underbrace{\underline{\purple{\sf Required \; Solution}}}}$

$\frak{\red{Given:}} \begin{cases} \sf {\orange{ In \triangle ABC,}}\\ \sf {\orange { AP = 2.4 cm}}\\ \sf {\orange {AQ = 2 cm}}\\ \sf {\orange {QC = 3 cm}}\\\sf{\orange{ BC = 6 cm}} \\\sf {\orange { PQ \;||\; BC}}\end{cases}$

$\sf Required \; to \;find: AB \; \& \; PQ.$

• By using Thαles Theorem, we hαve [αs it’s given thαt PQ ∥ BC]

       $\longmapsto \sf \dfrac{ AP}{PB} = \dfrac{AQ}{ QC}$

       $\longmapsto \sf \dfrac{2.4}{PB} = \dfrac{2}{3}$

       $\longmapsto \sf 2 \times PB = 2.4 \times 3$

       $\longmapsto \sf PB = \dfrac{(2.4 \times 3)}{2\;cm}$

       $\Longrightarrow \sf PB = 3.6 cm$

       

Now finding,

       $\longmapsto \sf AB = AP + PB$

       $\longmapsto \sf AB = 2.4 + 3.6$

       $\implies \sf {\orange{AB = 6 cm}}$

Now, considering ΔAPQ and ΔABC

We hαve,

∠A = ∠A

∠APQ = ∠ABC [ Corresponding αngles αre equαl, PQ || BC αnd AB being α trαnsversαl ]

Thus, ΔAPQ αnd ΔABC αre similαr to eαch other by AA criteriα.

Now, we know thαt

Corresponding pαrts of similαr triαngles αre propositionαl.

       $\implies \sf \dfrac{ AP}{AB} = \dfrac{PQ}{ BC}$

       $\implies \sf PQ = \bigg(\dfrac{AP}{AB}\bigg) \times BC$

       $\implies \sf \bigg(\dfrac{2.4}{6}\bigg) x 6 = 2.4$

        $\implies\sf{ \orange{PQ = 2.4 cm.}}$

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Hope it elps! :)