Question

What will be the new position of the given point (6, -5) after rotating 90 degrees counterclockwise about the origin

Answer 1

Answer:

the new position of the point will be ($5\sqrt{61} ,6\sqrt{61}$)

Step-by-step explanation:

Please see the image for visualisation.

when we rotate any point A(x,y) counterclockwise about origin by some angle $\theta$, then the coordinates of the new point B will change but the distance of A from origin would be the same as the distance from B.

i.e.  |OA| = |OB| = $\sqrt{6^{2}+(-5)^{2} }$=$\sqrt{61}$.

Thus, by using trigonometry we can write the coordinates of B as

($\sqrt{61}cos \alpha$ , $\sqrt{61}sin\alpha$) where $\alpha$ is the angle that OB makes with positive x-axis.

Since OAB is a right angle triangle , AB=$\sqrt{122}$.

now , use distance formula to find the distance between A & B where use the coordinates of B as ($\sqrt{61}cos \alpha$ , $\sqrt{61}sin\alpha$) and A as (6,-5).

Equate this equation to $\sqrt{122}$.

Simplify and you will get that 12$\sqrt{61}$cos$\alpha$ = 10$\sqrt{61}$sin$\alpha$.

thus, Tan$\alpha$= $\frac{6}{5}$. now, use the image and you will see that sin$\alpha$=6 and cos$\alpha$=5.

Put these values in ($\sqrt{61}cos \alpha$ , $\sqrt{61}sin\alpha$) to get coordinates of B.

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