In the given figure if LM || AB, AL = x – 3, AC = 2x,BM = x – 2 an

1.	In the given figure if LM \|\| AB, AL = x – 3, AC = 2x,BM = x – 2 and BC = 2x + 3. Then the value of x is …………….(A) 7 (B) 8 (C) 9 (D) Cannot be determined
Answer» Correct option is (C) 9 \(\because\) LM \|\| AB \(\therefore\) \(\angle CLM=\angle CAB\) (Corresponding angles as LM \|\| AB) & \(\angle CML=\angle CBA\) (Corresponding angles as LM \|\| AB) \(\therefore\) \(\triangle CLM\sim\triangle CAB\) (By AA similarity rule) \(\therefore\) \(\frac{LC}{AC}=\frac{MC}{BC}\) (By properties of similar triangles) \(\Rightarrow\) \(\frac{AC-AL}{AC}=\frac{BC-BM}{BC}\) \(\Rightarrow\) \(\frac{2x-(x-3)}{2x}=\frac{2x+3-(x-2)}{2x+3}\) (Given) \(\Rightarrow\) \(\frac{x+3}{2x}=\frac{x+5}{2x+3}\) \(\Rightarrow(x+3)(2x+3)=2x(x+5)\) \(\Rightarrow2x^2+9x+9=2x^2+10x\) \(\Rightarrow10x-9x=9\) \(\Rightarrow x=9\) Correct option is: (C) 9

In the given figure if LM || AB, AL = x – 3, AC = 2x,BM = x – 2 and BC = 2x + 3. Then the value of x is …………….(A) 7 (B) 8 (C) 9 (D) Cannot be determined

Answer»

Correct option is (C) 9

\(\because\) LM || AB

\(\therefore\) \(\angle CLM=\angle CAB\) (Corresponding angles as LM || AB)

& \(\angle CML=\angle CBA\) (Corresponding angles as LM || AB)

\(\therefore\) \(\triangle CLM\sim\triangle CAB\) (By AA similarity rule)

\(\therefore\) \(\frac{LC}{AC}=\frac{MC}{BC}\) (By properties of similar triangles)

\(\Rightarrow\) \(\frac{AC-AL}{AC}=\frac{BC-BM}{BC}\)

\(\Rightarrow\) \(\frac{2x-(x-3)}{2x}=\frac{2x+3-(x-2)}{2x+3}\) (Given)

\(\Rightarrow\) \(\frac{x+3}{2x}=\frac{x+5}{2x+3}\)

\(\Rightarrow(x+3)(2x+3)=2x(x+5)\)

\(\Rightarrow2x^2+9x+9=2x^2+10x\)

\(\Rightarrow10x-9x=9\)

\(\Rightarrow x=9\)

Correct option is: (C) 9