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

If the green triangle π΄πΆπ΅ is rotated about a point π with an angle of 72Β°, which triangle represents its final position?

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Video Transcript

If the green triangle π΄πΆπ΅ is rotated about a point π with an angle of 72 degrees, which triangle represents its final position? (A) Triangle πΊπ»πΌ, (B) triangle πΌπ½π΄, (C) triangle πΈπΉπΊ, or (D) triangle π΅π·πΈ.

We are looking for the image of triangle π΄πΆπ΅ following a rotation of 72 degrees about point π. 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 π΄, πΆ, and π΅.

We recall that we rotate a point π about π΄ by moving it along a circle centered at π΄ with radius π΄π. And the measure of angle ππ΄π prime is equal to the measure of the rotation. We want to determine the images of the vertices of triangle π΄πΆπ΅. So letβs start with π΄.

Weβll begin by sketching a circle centered at π of radius ππ΄. We note that points π΅, πΈ, πΊ, and πΌ also lie on this circle. Thus, we have radii ππ΅, ππΈ, ππΊ, and ππΌ. We also note that polygon π΄π΅πΈπΊπΌ is a regular pentagon. So the angles around π are all congruent. We know that the sum of the angles around π 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 π΄ after rotating 72 degrees counterclockwise coincides with π΅. Similarly, rotating vertex π΅ 72 degrees counterclockwise coincides with point πΈ.

We can follow the same process for vertex πΆ. We begin by sketching a circle centered at π with radius ππΆ. Then, we can sketch four more radii found by connecting π to the other points on the circle of radius ππΆ. Using the same reasoning as before, we conclude that each central angle has a measure of 72 degrees. Thus, the 72-degree rotation of πΆ along the circle coincides with point π·. So the image of π΄ is π΅, the image of πΆ is π·, and the image of π΅ is πΈ. Hence, the image of triangle π΄πΆπ΅ is triangle π΅π·πΈ, which is answer (D).