In a ΔABC , A = (1,2) ; B =(5,5) In angleACB = 90° If area of ΔABC is to be 6.5 squnits the possible number of points for C are <

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1. Topic :-

   Coordinate Geometry

   Given :-

   In a ΔABC, A ≡ (1, 2); B ≡ (5, 5) and ∠ACB = 90°.

   Area of ΔABC is to be 6.50 sq. units.

   To Find :-

   Possible number of points for C.

   Solution :-

   ΔABC is a right angle triangle as ∠ACB = 90°.

   Side AB opposite to ∠ACB will act as hypotenuse for the triangle.

   Calculating length AB from Distance Formula,

   [tex]AB=\sqrt{(5-1)^2+(5-2)^2}\;units[/tex]

   [tex]AB=\sqrt{4^2+3^2}\;units[/tex]

   [tex]AB=\sqrt{16+9}\;units[/tex]

   [tex]AB=\sqrt{25}\;units[/tex]

   [tex]AB=5\;units[/tex]

   Assuming length of arms of triangle,

   Let length of arms of given triangle be x and y.

   Applying Pythagoras Theorem,

   x² + y² = 5² . . . . equation (1)

   Area of Right Angle Triangle,

   [tex]Area=\dfrac{1}{2} \times (Product\:of\:length\:of\:arms)[/tex]

   [tex]6.5=\dfrac{1}{2} \times x \times y[/tex]

   [tex]13=xy[/tex]

   [tex]x=\dfrac{13}{y}[/tex]

   Substituting value of ‘x’ in equation (1),

   [tex]x^2+y^2=5^2[/tex]

   [tex]\left( \dfrac{13}{y} \right)^2+y^2=5^2[/tex]

   [tex]\dfrac{169}{y^2} +y^2=25[/tex]

   [tex]\dfrac{169+y^4}{y^2}=25[/tex]

   Cross Multiply,

   [tex]169+y^4=25y^2[/tex]

   Rearranging it,

   [tex]y^4-25y^2+169=0[/tex]

   Substitite y² = t,

   [tex](y^2)^2-25y^2+169=0[/tex]

   [tex]t^2-25t+169=0[/tex]

   Calculating value of Discriminant,

   [tex]D=b^2-4ac[/tex]

   Here,

   a = 1

   b = -25

   c = 169

   [tex]D=(-25)^2-4(1)(169)[/tex]

   [tex]D=625-676[/tex]

   [tex]D=-51[/tex]

   [tex]D<0[/tex]

   which means

   Real ‘t’ doesn’t exist which means Real ‘y’ doesn’t exist.

   Thus, there are no possible point for point C as y doesn’t exist.

   Answer :-

   So, there are Zero (0) possible points for point C.

   Hence, option C is correct.

   Reply

2. So, there are Zero (0) possible points for point C.

   Hence, option C is correct.

   0 is your answer.

   Hope it will help you..

   Reply