Question : If the altitude of a triangle is increased by 10% while its area remains the same, its corresponding base will have to be decreased by:
Option 1: $10\%$
Option 2: $9\%$
Option 3: $9\frac{1}{11}\%$
Option 4: $11\frac{1}{9}\%$
Correct Answer: $9\frac{1}{11}\%$
Solution :
Given that the area remains the same and the altitude is increased by $10\%$, the base must decrease to compensate for this.
Let the original base be $B_o$ and the original altitude be $A_o$.
The new altitude is $1.1A_n$.
Let the new base be $B_n$.
$⇒\frac{1}{2} \times B_o \times A_o= \frac{1}{2} \times B_n \times 1.1A_n$
$⇒B_n = \frac{B_o}{1.1}$
The percentage decrease in base $=(\frac{B_o-B_n}{B_o})×100$
The percentage decrease in base $=(\frac{B_o-\frac{B_o}{1.1}}{B_o})×100$
The percentage decrease in base $=(1-\frac{1}{1.1})×100=9\frac{1}{11}\%$
Hence, the correct answer is $9\frac{1}{11}\%$.
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