Video: Using the Axes of Symmetry of a Shape to Solve a Problem

Segment 𝐢𝐷 has mirror symmetry in line 𝐴𝐹. Given that 𝐸𝐷 = 5 and 𝐡𝐢 = 5.1, calculate the perimeters of 𝐴𝐢𝐹𝐷 and △𝐡𝐢𝐷.

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Video Transcript

Segment 𝐢𝐷 has mirror symmetry in the line 𝐴𝐹. Given that 𝐸𝐷 is equal to five and 𝐡𝐢 is equal to 5.1, calculate the perimeters of 𝐴𝐢𝐹𝐷 and triangle 𝐡𝐢𝐷.

As the line 𝐴𝐹 is a line of symmetry, we know that the lengths of several sides will be equal. The length 𝐴𝐢 will be equal to the length 𝐴𝐷. Both of these have length 6.7. The lengths 𝐡𝐢 and 𝐡𝐷 are also equal in length. We are told that 𝐡𝐢 is equal to 5.1. Therefore, 𝐡𝐷 is also equal to 5.1. The lengths 𝐸𝐷 and 𝐸𝐢 are also equal in length. 𝐸𝐷 is equal to five. Therefore, 𝐸𝐢 is also equal to five. Finally, the lengths 𝐷𝐹 and 𝐢𝐹 are also equal. We’re told in the diagram that 𝐢𝐹 is equal to 8.2. Therefore, 𝐷𝐹 is also equal to 8.2.

The first perimeter we’re asked to calculate is that of 𝐴𝐢𝐹𝐷. This is the outside of the shape. We can calculate this by adding the four lengths 𝐴𝐢, 𝐢𝐹, 𝐹𝐷 and 𝐷𝐴. 𝐴𝐢 and 𝐴𝐷 or 𝐷𝐴 are both equal to 6.7. 𝐢𝐹 and 𝐹𝐷 or 𝐷𝐹 are both equal to 8.2. We need to add 6.7, 8.2, 8.2, and 6.7. 6.7 plus 8.2 is equal to 14.9. This means that 8.2 plus 6.7 is also equal to 14.9. 14.9 plus 14.9 or two times 14.9 is equal to 29.8. The perimeter of 𝐴𝐢𝐹𝐷 is 29.8.

We also need to calculate the perimeter of the triangle 𝐡𝐢𝐷. This is equal to the three lengths 𝐡𝐢, 𝐢𝐷 and 𝐷𝐡. This can be split into four lengths that we know, 𝐡𝐢, 𝐢𝐸, 𝐸𝐷 and 𝐷𝐡. 𝐡𝐢 and 𝐡𝐷 are both equal to 5.1. 𝐢𝐸 and 𝐸𝐷 are both equal to five. We need to add 5.1, five, five, and 5.1. 5.1 plus five is equal to 10.1. We get the same answer when we add them the other way round. 10.1 plus 10.1 is equal to 20.2. The perimeter of triangle 𝐡𝐢𝐷 is 20.2. Therefore, our two correct answers are 29.8 and 20.2.

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