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I am unable to solve this question.
If the area of a rhombus is 10 sq.unit . It's diagonals intersect at (0,0) if one vertex of the rhombus is (3,4) , then one of the other vertices can be ?
I took a rhombus as ABCD . I took A as (3,4) and took O(0,0) as the point of the intersections of the diagonals . I found out OA as 5 and OB as 1 . I found out C as(-3,-4) . Now , the problem is that I am unable to find the vertex B . Please tell me how do I find out the vertex B . Thank you!

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  • $\begingroup$ Hint: vectors $(a,b)$ and $(-b,a)$ are always orthogonal. $\endgroup$ May 25 '15 at 11:40
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The length of $A C$ is 10, so the length of $B D$ is 2. Now $O C$ and $O B$ are orthogonal, so $B = t (4, -3)$. Putting in the distance you have $t = 1/5$.

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  • $\begingroup$ Thank you ! I wanted to ask you how can we say that OC and OB are orthogonal and why have we put t=1/5 .Thank you! $\endgroup$
    – Seeker1201
    May 25 '15 at 11:46
  • $\begingroup$ OC and OB are orthogonal by definition of the rhombus (use Euclidean geometry and isosceles triangles to prove if you really want to), $|B|=1$, solve that and you get $t$. $\endgroup$
    – simonzack
    May 25 '15 at 11:47
  • $\begingroup$ Thank you ! Is this the reason we have taken (4,-3) as points ? Thank you ! $\endgroup$
    – Seeker1201
    May 25 '15 at 11:56
  • $\begingroup$ @Nancy Yes, as Alexey also mentions this is required for orthogonality. $\endgroup$
    – simonzack
    May 25 '15 at 11:59
  • $\begingroup$ Thank you . My book has taken the points as (-4,3) . Can you tell me why ? $\endgroup$
    – Seeker1201
    May 25 '15 at 12:11
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As we know that the diagonals of rhombus are equal in length & intersect each other normally. Distance of each vertex from the origin is $\sqrt{3^2+4^2}=5$.

Thus assuming the rhombus ABCD, the vertex C opposite to $A\equiv (3, 4)$ can be easily determined as the moi-point of AC is $(0, 0)$ Hence, $C(-3, -4)$. Now, assume any vertex say $(x, y)$ on the other diagonal BD. Since, the semi-diagonal OB is normal to OA, we get $$\left(\frac{y-0}{x-0}\right)\left(\frac{4-0}{3-0}\right)=-1 \implies y=-\frac{3}{4}x \tag 1$$ Since $OB=OA=\sqrt{3^2+4^2}=5$, we get $$\sqrt{(x-0)^2+(y-0)^2}=5 \implies x^2+y^2=25$$ Setting the value of $y$, we get $$x^2+\left(-\frac{3}{4}x\right)^2=25 \implies x^2=16 \implies x=\pm 4$$$$ x=4\quad \implies y=-\frac{3}{4}(4)=-3$$ $$ x=-4\quad \implies y=-\frac{3}{4}(-4)=3$$ Thus, all the unknown vertices of rhombus ABCD are $B\equiv (4, -3)$, $C\equiv(-3, -4)$ & $D\equiv(-4, 3)$

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