Question Video: Rotating a Triangle about a Point Mathematics • 8th Grade

Video Transcript

When the green triangle is rotated 90 degrees about 𝐴, which triangle is its new position? (A) Triangle 𝐴𝐷𝐸, (B) triangle 𝐴𝐹𝐺, or (C) triangle 𝐴𝐻𝐼.

We are asked to rotate triangle 𝐴𝐵𝐶 90 degrees about point 𝐴. We recall that to rotate a polygon about a point, we first rotate all its vertices. So, we’ll need to rotate points 𝐴, 𝐵, and 𝐶 90 degrees. Since 90 degrees is a positive value, we note that we will rotate all these points counterclockwise.

We recall that we rotate a point 𝑋 about 𝐴 by moving it along a circle centered at 𝐴 with radius 𝐴𝑋. And the measure of angle 𝑋𝐴𝑋 prime is equal to the measure of the rotation, where 𝑋 prime is the image of 𝑋. We will follow this procedure to find the image of each vertex of the green triangle.

Let’s start with 𝐴. 𝐴 is the center of rotation, so it will not move under this rotation. Next, we’ll find the image of vertex 𝐵. We can draw a ray with endpoint 𝐴 passing through 𝐵 and then measure another ray that is 90 degrees in the counterclockwise direction of ray 𝐴𝐵. Let’s name this ray 𝐴𝐽. If we then sketch a circle centered at 𝐴 with radius 𝐴𝐵, we note that 𝐷 is the point of intersection between the circle and ray 𝐴𝐽 and that the measure of angle 𝐵𝐴𝐷 is 90 degrees. Thus, the image of 𝐵 after the rotation is 𝐷.

We can follow the same process for vertex 𝐶. First, we sketch the ray with endpoint 𝐴 passing through point 𝐶 and add another ray that is 90 degrees counterclockwise from ray 𝐴𝐶. We can then sketch a circle of radius 𝐴𝐶 centered at 𝐴 and find that the point of intersection between the ray and the circle is point 𝐸.

So, the image of 𝐴 is 𝐴, the image of 𝐵 is 𝐷, and the image of 𝐶 is 𝐸. Therefore, the image of triangle 𝐴𝐵𝐶 is triangle 𝐴𝐷𝐸, which is answer (A).