question_answer

In the given figure, ABC is an equilateral triangle of side length 30 cm XY is parallel to BC, XP s parallel to AC and YQ is parallel to AB. If\[(XY+XP-YQ)\] is 40 cm, then what is PQ equal to

A) 5 cm    

B)  12 cm  

C)  15 cm

D)  10 cm

Correct Answer: D

Solution :

Since, XP||AC, YQ||AB

\[\therefore \]      \[\angle XBP=\angle YQC\] and \[\angle XPB=\angle YCQ\]

\[\therefore \]\[\Delta XBP\]and \[\Delta YCQ\]are equilateral triangles.

Now, XY|| BC

\[\therefore \]      \[\frac{AX}{AB}=\frac{XY}{BC}\]

\[\Rightarrow \]   \[AX=XY\]        \[(\because AB=BC=30\,\,\text{cm})\]

Also,     \[XY+XP+YQ=40\]

\[\Rightarrow \]\[AX+XB+YQ=40\]  \[(\because XY=AX,XP=XB)\]

\[\Rightarrow \]   \[AB+YQ=40\]

\[\Rightarrow \]   \[YQ=40-30=10\,\,\text{cm}\]

\[\therefore \]      \[YQ=XP=10\,\,\text{cm}\]

\[\therefore \]      \[BP=CQ=10\,\,\text{cm}\]

\[\therefore \] \[PQ=30-BP-CQ=30-10-10=10\,\,\text{cm}\]