Question

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?

Answer 1

`(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.!

Answer 2

`(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.!