Question Video: Rotating a Line Segment about a Point | Nagwa Question Video: Rotating a Line Segment about a Point | Nagwa

Question Video: Rotating a Line Segment about a Point Mathematics • First Year of Preparatory School

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If line segment 𝐴𝐡 is rotated about 𝑀 with a 90Β° angle, which line segment represents its final position?

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Video Transcript

If line segment 𝐴𝐡 is rotated about 𝑀 with a 90-degree angle, which line segment represents its final position? (A) Line segment 𝐷𝐴, (B) line segment 𝐢𝐷, or (C) line segment 𝐡𝐢.

We want to rotate line segment 𝐴𝐡 90 degrees about point 𝑀. We recall that rotating by a positive value is rotating in the counterclockwise direction. To rotate a line segment, we recall that we can just rotate the endpoints. In other words, the image of line segment 𝐴𝐡 would be line segment 𝐴 prime 𝐡 prime.

Let’s start by finding the image of 𝐴. We want to rotate point 𝐴 90 degrees about point 𝑀 by finding point 𝐴 prime, which is counterclockwise on the circle shown here centered at 𝑀 of radius 𝐴𝑀 such that the measure of angle 𝐴𝑀𝐴 prime is equal to 90 degrees.

We can do this by noticing that the distance between 𝑀 and any vertex of the square is the same. So, all the vertices lie on the circle of radius 𝐴𝑀 centered at 𝑀. Then, we can measure the angles at the center of the square by noting that these angles are congruent and their measures sum to 360 degrees. So, each of the four angles at the center measure 90 degrees. Thus, the measure of angle 𝐴𝑀𝐡 is equal to 90 degrees. And we see that 𝐴 prime is equal to 𝐡. Then, we can follow the same reasoning to show that 𝐡 prime is equal to 𝐢.

Hence, the image of line segment 𝐴𝐡 after a rotation of 90 degrees is line segment 𝐡𝐢, which is answer (C).

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