An isosceles triangle has sides A, B, and C with sides B and C being equal in length. If side A goes from #(4 ,9 )# to #(8 ,5 )# and the triangle's area is #34 #, what are the possible coordinates of the triangle's third corner?

2 Answers
Ratnaker Mehta
Mar 4, 2017

#(29/2,31/2) or, (-5/2,-3/2).#

Explanation:

We have #2# Methods to solve this Problem.

#1^(st)# Method :-

Let #Q(4,9) and R(8,5).#

#A=QR=sqrt{(4-8)^2+(9-5)^2}=4sqrt2.#

#"Area of "Delta=1/2"(Base)(Altitude)"#

#:. 34=1/2(A)"(Altitude)"rArr"Altitude="{(34)(2)}/(4sqrt2)=17/sqrt2.#

This means the the #3^(rd) "Vertex"#, call it #P(x,y),# is situated at a

#bot-"distance"# of #17/sqrt2# from the line-sgmt. #QR=A.#

At the same, #Delta PQR" is isisceles with, "PQ [=B=C] = PR.#

Therefore, #P# lies on the #bot-#bisector of #QR#, at a distance of

#17/sqrt2.#

Let #M# is the mid-point of #QR.#

#:. M=M(6,7), PM=17/sqrt2, and, PM bot QR.#

Slope of #QR=-1 rArr "Slope of "PM=1=tan(pi/4).#

Now, to find #P,#we use the following very useful Result:-

Result :- On a line #l,# passing through #(x_0,y_0)# & making an

angle #theta# with the #+ve# X-axis, the point on #l# at a dist.

#r# from #(x_0,y_0)# is given by #(x_0+-rcostheta,y_0+-rsintheta).#

Here, we have #l=PM, (x_0,y_0)=M(6,7), theta=pi/4, and, r=PM=17/sqrt2.#

#:.," for "P(x,y), x=6+-(17/sqrt2)cos(pi/4)#

#=6+-(17/sqrt2)(1/sqrt2)#

#:. x=29/2, or, -5/2.#

Similarly, #y=31/2, or, -3/2.#

Hence, the possible co-ords. of the #3^(rd)# vertex are #(29/2,31/2)#

or, #(-5/2,-3/2).#

Enjoy Maths.!

Ratnaker Mehta
Mar 4, 2017

# (29/2,31/2), or, (-5/2,-3/2).#

Explanation:

#2^(nd)# Method :-

Let #Q(4,9) and R(8,5)# and let the #3^(rd)# vertex be #P(x,y).#

Now, the Area of #DeltaPQR=34#

#rArr 1/2|D|=34 rArr , where,#,

#D=|(x,y,1),(4,9,1),(8,5,1)|=|4x+4y-52|=4|x+y-13|.#

#:. 2|x+y-13|=34#

#:. x+y-13=+-17.#

Let, #x+y=30............(1'), or, x+y=-4..............(1'').#

Next, we have, #PQ=PR#

#:.(x-4)^2+(y-9)^2=(x-8)^2+(y-5)^2.#

#:. 8x-8y=-8, or, x-y=-1...................(2)#.

Solving #(1') and (2), x=29/2, y=31/2; and,#

#(1'') and (2) rArr x=-5/2, y=-3/2.#

#:. P(x,y)=(29/2,31/2), or, (-5/2,-3/2),# as we had in the

#1^(st)# Method.

Enjoy Maths.!

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