Question

select the correct answer.
a quadrilateral in the third quadrant is rotated 90° clockwise about the origin. which quadrant will the transformed quadrilateral be in?
a. first quadrant
b. second quadrant
c. third quadrant
d. fourth quadrant

Explanation:

Step1: Recall quadrant properties

In the third quadrant, both \(x\) and \(y\) coordinates of a point are negative (e.g., \((-a, -b)\) where \(a,b>0\)).

Step2: Apply 90° clockwise rotation rule

The rule for rotating a point \((x, y)\) 90° clockwise about the origin is \((x, y)\to(y, -x)\). For a point \((-a, -b)\) (third quadrant), applying the rule: \(x=-a\), \(y = -b\), so the new point is \((-b, a)\) (since \(-x=-(-a)=a\)).

Step3: Determine new quadrant

In the point \((-b, a)\), \(x=-b<0\) and \(y = a>0\). Points with \(x<0\) and \(y>0\) lie in the second quadrant? Wait, no—wait, wait, let's re - check. Wait, original point in third quadrant: \((x,y)=(-a,-b)\), \(a,b>0\). Rotating 90° clockwise: \((x,y)\to(y, -x)\), so \((-b, -(-a))=(-b,a)\). Wait, \(x=-b\) (negative), \(y = a\) (positive). Wait, no, that's second quadrant? Wait, no, maybe I made a mistake. Wait, let's take a concrete example. Let the point be \((-1,-1)\) (third quadrant). Rotating 90° clockwise: \((-1,-1)\to(-1,1)\)? Wait, no, the rule is \((x,y)\to(y, -x)\). So for \((-1,-1)\), \(x=-1\), \(y=-1\), so new point is \((-1, -(-1))=(-1,1)\). Wait, \((-1,1)\) is in the second quadrant? But that contradicts. Wait, no, wait another example: point \((-2,-3)\) (third quadrant). Rotating 90° clockwise: \((-3, 2)\) (since \(y=-3\), \(-x = 2\)). So \((-3,2)\): \(x=-3<0\), \(y = 2>0\) → second quadrant? Wait, but the options have B as second quadrant? Wait, no, maybe I messed up the rotation direction. Wait, 90° clockwise rotation: the standard rule is \((x,y)\to(y, -x)\). Let's take a point in first quadrant \((1,2)\), rotate 90° clockwise: \((2,-1)\) (fourth quadrant). Point in second quadrant \((-1,2)\), rotate 90° clockwise: \((2,1)\) (first quadrant). Point in third quadrant \((-1,-2)\), rotate 90° clockwise: \((-2,1)\)? Wait, no, \((x,y)=(-1,-2)\), so \(y=-2\), \(-x = 1\), so \((-2,1)\)? Wait, \(x=-2<0\), \(y = 1>0\) → second quadrant. Wait, but let's check with the coordinate plane. Third quadrant is bottom - left. Rotating 90° clockwise: imagine turning the point 90° clockwise around the origin. A point in third quadrant, when rotated 90° clockwise, will move to the second quadrant? Wait, no, maybe I had the rotation rule wrong. Wait, 90° counter - clockwise is \((x,y)\to(-y,x)\), 90° clockwise is \((x,y)\to(y, -x)\). Let's use the unit circle. A point in third quadrant is at an angle between \(180^{\circ}\) and \(270^{\circ}\). Rotating 90° clockwise (subtracting \(90^{\circ}\) from the angle), so the new angle is between \(90^{\circ}\) and \(180^{\circ}\), which is the second quadrant. So the transformed quadrilateral (all its points will follow this) will be in the second quadrant.

Answer:

B. second quadrant