Question

Three horses are tethered with 7-meter-long ropes at the three corners of a triangular field having sides 20 m, 34 m and 42 m. Find the area of the plot which can be grazed by the horses and the area of the plot which remains ungrazed.

• A. $77m^2;259m^2$
• B. $68m^2;255m^2$
• C. $66m^2;257m^2$
• D. $77m^2;269m^2$

Answer 1

The correct option is A $77m^2;259m^2$

Given,
Sides of triangular field is $20m,34mand42m$
Semi-perimeter = $\frac{20m+34m+42m}{2}$
=$\frac{96}{2}$
=$48m$
Area of field=$\sqrt{48(48-20)(48-34)(48-42)}$
=$\sqrt{48\times 28\times 14\times 6}$
=$\sqrt{112896}$
=$336m^2$
We know that sum of angles of triangles =$180$
Thus, Area gazed
=Area of sector APQ+Area of sector BRS+Area of sector CTU
=Area of semicircle with radius $7$m
=$\frac{\pi }{2}\times (7m{)}^2$
=$\frac{22}{14}\times (49m{)}^2$
=$77m^2$
Area of field-Area of gazed
$=(336-77)m^2$
$=259m^2$
Area of ungazed is $259m^2$