A triangle #hat(ABC)# has vertices of #A(1,3);B(1/2,3/2);C(2,1)#. Verify that the triangle is isosceles and calculate the area and perimeter?

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Answer 1 · 2017-07-07T13:53:38

#" Area="5/4," the Perimeter="sqrt5(1+sqrt2).#

Let us first verify that the given points are non-collinear.

We use the following necessary and sufficient condition for

collinearity of the points :

#A(x_1,y_1), B(x_2,y_2) and C(x_3,y_3)" are collinear "iff #

#|(x_1,y_1,1),(x_2,y_2,1),(x_3,y_3,1)|=0.#

We have, #D=|(1,3,1),(1/2,3/2,1),(2,1,1)|,#

#=1(3/2-1)-3(1/2-2)+1(1/2-3),#

#=1/2+9/2-5/2=5/2.#

Thus, the points are not collinear, and, hence, form #DeltaABC.#

Knowing that, the Area of #DeltaABC# is #1/2|D|,#

The Reqd. Area =#5/4.#

Using the Distance Formula, we have,

#AB^2=(1-1/2)^2+(3-3/2)^2=1/4+9/4 rArr AB=sqrt10/2.#

#BC^2=(1/2-2)^2+(3/2-1)^1=9/4+1/4 rArr BC=sqrt10/2.#

#AC^2=(1-2)^2+(3-1)^2=1+4 rArr AC=sqrt5.#

#because," in "DeltaABC, AB=BC, :., Delta "is isisceles,"# having

#"perimeter="AB+BC+AC=sqrt5+sqrt10=sqrt5(1+sqrt2).#

Enjoy Maths.!