Question Video: Using the Axes of Symmetry of a Shape to Solve a Problem Mathematics • 8th Grade

Video Transcript

Segment 𝐶𝐷 has mirror symmetry in the line 𝐴𝐹. Given that 𝐸𝐷 is equal to five and 𝐵𝐶 is equal to 5.1, calculate the perimeters of 𝐴𝐶𝐹𝐷 and triangle 𝐵𝐶𝐷.

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

The first perimeter we’re asked to calculate is that of 𝐴𝐶𝐹𝐷. This is the outside of the shape. We can calculate this by adding the four lengths 𝐴𝐶, 𝐶𝐹, 𝐹𝐷 and 𝐷𝐴. 𝐴𝐶 and 𝐴𝐷 or 𝐷𝐴 are both equal to 6.7. 𝐶𝐹 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 𝐴𝐶𝐹𝐷 is 29.8.

We also need to calculate the perimeter of the triangle 𝐵𝐶𝐷. This is equal to the three lengths 𝐵𝐶, 𝐶𝐷 and 𝐷𝐵. This can be split into four lengths that we know, 𝐵𝐶, 𝐶𝐸, 𝐸𝐷 and 𝐷𝐵. 𝐵𝐶 and 𝐵𝐷 are both equal to 5.1. 𝐶𝐸 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 𝐵𝐶𝐷 is 20.2. Therefore, our two correct answers are 29.8 and 20.2.