Question
In $\triangle\text{ABC and }\triangle\text{DEF},$ it is being given that: AB = 5cm, BC = 4cm and CA = 4.2cm; DE = 10cm, EF = 8cm and FD = 8.4cm. If $\text{AL}\perp\text{BC}$ and $\text{DM} \perp \text{EF,}$ find AL : DM.
✓
Answer
Since, $\frac{\text{AB}}{\text{DE}}=\frac{\text{BC}}{\text{EF}}=\frac{\text{AC}}{\text{DE}}=\frac{1}{2}$Then, $\triangle\text{ABC}\sim\triangle\text{DEF}$ [By SSS similarity]
Now, In $\triangle\text{ABL}\sim\triangle\text{DEF}$
$\angle\text{B}=\angle\text{E}$ $\big[\triangle\text{ABC}\sim\triangle\text{DEF}\big]$
$\angle\text{ALB}=\angle\text{DME}$ [Each 90°]
Then, $\triangle\text{ABL}\sim\triangle\text{DEM}$ [By AA similarity]
$\therefore\frac{\text{AB}}{\text{DE}}=\frac{\text{AL}}{\text{DM}}$ [Corresponding parts of similar $\triangle$ are proportional]
$\Rightarrow\frac{5}{10}=\frac{\text{AL}}{\text{DM}}$
$\Rightarrow\frac{1}{2}=\frac{\text{AL}}{\text{DM}}$
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