(Numbers indicate the lengths of the sides of the triangles) If the area of (bigtriangleup) PQR is k2sq. units what is the area of the shades portion?
-
A.
(frac{5}{9})k2 sq. units -
B.
(frac{1}{3})k2 sq. units -
C.
(frac{8}{9})k2 sq. units -
D.
(frac{7}{9})k2 sq. units -
E.
(frac{2}{3})k2 sq. u
Correct Answer: Option A
Explanation
Area of shaded portion = Area of triangle PQR – Area of inner triangle
Area of triangle given 3 sides a, b, c = (sqrt{s(s – a)(s – b)(s – c)})
where (s = frac{a + b + c}{2} )
Area of PQR :
(s = frac{3 + 5 + 6}{2} = frac{14}{2} = 7)
Area = (sqrt{7(7 – 3)(7 – 5)(7 – 6)})
= (sqrt{7(4)(2)(1)} = sqrt{56})
(implies K^{2} = sqrt{56})
Area of inner triangle :
(s = frac{2 + 4 + frac{10}{3}}{2} = frac{14}{3})
Area = (sqrt{frac{14}{3} (frac{14}{3} – 2)(frac{14}{3} – 4)(frac{14}{3} – frac{10}{3})})
= (sqrt{frac{14}{3} (frac{8}{3})(frac{2}{3})(frac{4}{3})})
= (sqrt{frac{896}{81}})
= (sqrt{frac{16}{81}} times sqrt{56})
= (frac{4}{9} K^{2})
(therefore text{The area of the shaded portion} = K^{2} – frac{4}{9}K^{2} = frac{5}{9}K^{2})