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

If the green triangle 𝐴𝐢𝐡 is rotated about a point 𝑀 with an angle of 72Β°, which triangle represents its final position?

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Video Transcript

If the green triangle 𝐴𝐢𝐡 is rotated about a point 𝑀 with an angle of 72 degrees, which triangle represents its final position? (A) Triangle 𝐺𝐻𝐼, (B) triangle 𝐼𝐽𝐴, (C) triangle 𝐸𝐹𝐺, or (D) triangle 𝐡𝐷𝐸.

We are looking for the image of triangle 𝐴𝐢𝐡 following a rotation of 72 degrees about point 𝑀. We recall that to rotate a shape by a positive value is to rotate the shape counterclockwise. We rotate a triangle about a point by rotating all of its vertices. So we must find the image of each vertex 𝐴, 𝐢, and 𝐡.

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. We want to determine the images of the vertices of triangle 𝐴𝐢𝐡. So let’s start with 𝐴.

We’ll begin by sketching a circle centered at 𝑀 of radius 𝑀𝐴. We note that points 𝐡, 𝐸, 𝐺, and 𝐼 also lie on this circle. Thus, we have radii 𝑀𝐡, 𝑀𝐸, 𝑀𝐺, and 𝑀𝐼. We also note that polygon 𝐴𝐡𝐸𝐺𝐼 is a regular pentagon. So the angles around 𝑀 are all congruent. We know that the sum of the angles around 𝑀 must be 360 degrees. Thus, each angle at the center has a measure calculated by 360 degrees divided by five. That means each central angle has a measure of 72 degrees. So we see that the image of 𝐴 after rotating 72 degrees counterclockwise coincides with 𝐡. Similarly, rotating vertex 𝐡 72 degrees counterclockwise coincides with point 𝐸.

We can follow the same process for vertex 𝐢. We begin by sketching a circle centered at 𝑀 with radius 𝑀𝐢. Then, we can sketch four more radii found by connecting 𝑀 to the other points on the circle of radius 𝑀𝐢. Using the same reasoning as before, we conclude that each central angle has a measure of 72 degrees. Thus, the 72-degree rotation of 𝐢 along the circle coincides with point 𝐷. So the image of 𝐴 is 𝐡, the image of 𝐢 is 𝐷, and the image of 𝐡 is 𝐸. Hence, the image of triangle 𝐴𝐢𝐡 is triangle 𝐡𝐷𝐸, which is answer (D).

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