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Question Video: Identifying the Similarity between Two Rectangles given the Coordinates of Their Vertices Mathematics • 8th Grade

Consider the points 𝐴(3, 5), 𝐡(3, βˆ’5), 𝐢(5, βˆ’5), 𝐷(5, 6), π‘Š(βˆ’3, 8), 𝑋(βˆ’3, 28), π‘Œ(1, 28), and 𝑍(1, 8). Is the rectangle 𝐴𝐡𝐢𝐷 similar to the rectangle π‘Šπ‘‹π‘Œπ‘?

03:34

Video Transcript

Consider the points 𝐴 three, five; 𝐡 three, negative five; 𝐢 five, negative five; 𝐷 five, five; π‘Š negative three, eight; 𝑋 negative three, 28; π‘Œ one, 28; and 𝑍 one, eight. Is the rectangle 𝐴𝐡𝐢𝐷 similar to the rectangle π‘Šπ‘‹π‘Œπ‘?

We recall that for two shapes to be similar, they must be a dilation or enlargement of each other. This means that the dimensions of one shape must be multiplied by the same scale factor to find the dimensions of the second shape. In this question, we need to consider the two rectangles 𝐴𝐡𝐢𝐷 and π‘Šπ‘‹π‘Œπ‘. Noting that the sketch is not drawn to scale, the rectangle 𝐴𝐡𝐢𝐷 has vertices at three, five; three, negative five; five, negative five; and five, five.

The vertical distance between vertex 𝐴 and vertex 𝐡 is equal to 10, as five minus negative five equals 10. Likewise, the horizontal distance between vertices 𝐡 and 𝐢 and vertices 𝐴 and 𝐷 is equal to two, as five minus three is equal to two. The rectangle 𝐴𝐡𝐢𝐷 has dimensions equal to two units and 10 units.

We can repeat this process for the rectangle π‘Šπ‘‹π‘Œπ‘, once again noting that our diagram is not to scale. The vertical distance from vertex π‘Š to vertex 𝑋 is equal to 20 units, as 28 minus eight equals 20. The horizontal distance from π‘Š to 𝑍 or from 𝑋 to π‘Œ is equal to four, as one minus negative three equals four. This means that the rectangle π‘Šπ‘‹π‘Œπ‘ has dimensions equal to four units and 20 units.

If we consider our scale factor as going from the smaller rectangle 𝐴𝐡𝐢𝐷 to the larger rectangle π‘Šπ‘‹π‘Œπ‘, then for the vertical length, 10 units multiplied by our scale factor must equal 20 units. Dividing both sides of this equation by 10 gives us a scale factor of two. The length π‘Šπ‘‹ is twice the length of 𝐴𝐡.

We can then repeat this process for the lengths 𝐡𝐢 and π‘Šπ‘. Two multiplied by our scale factor will be equal to four. Dividing both sides of this equation by two, we once again get a scale factor equal to two. The rectangle π‘Šπ‘‹π‘Œπ‘ is therefore a dilation or enlargement of scale factor two of the rectangle 𝐴𝐡𝐢𝐷. We can therefore conclude that the two rectangles are similar. The correct answer is yes.

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