Given that the rectangle 𝐴𝐵𝐶𝐷 is similar to the rectangle 𝑋𝐵𝑍𝑌, find the length of line segment 𝑋𝑍.
We are told in the question that the rectangles 𝐴𝐵𝐶𝐷 and 𝑋𝐵𝑍𝑌 are similar. This means that they’re enlargements or dilations of each other. The corresponding lengths in each rectangle will, therefore, be in the same ratio. The ratio of side lengths 𝑋𝑌 and 𝐴𝐷 will be equal to the ratio of side lengths 𝑍𝑌 and 𝐶𝐷. Substituting the lengths in centimeters from our diagram, we have 𝑋𝑌 over 80 is equal to 21 over 84. The right-hand side simplifies to one-quarter by dividing the numerator and denominator by 21. We can then multiply both sides of this equation by 80. One-quarter multiplied by 80 is equal to 20. This means that the line segment 𝑋𝑌 is equal to 20 centimeters.
We can now use the smaller rectangle to find the length of 𝑋𝑍. The Pythagorean theorem states that, in any right triangle, 𝑎 squared plus 𝑏 squared is equal to 𝑐 squared, where 𝑎 and 𝑏 are the lengths of the shorter sides of the right triangle and 𝑐 is the length of the hypotenuse. In the right triangle 𝑋𝑌𝑍, 𝑋𝑍 is the hypotenuse. Therefore, 𝑋𝑌 squared plus 𝑌𝑍 squared is equal to 𝑋𝑍 squared. Substituting in the lengths of 𝑋𝑌 and 𝑌𝑍, the left-hand side becomes 20 squared plus 21 squared. 20 squared is equal to 400, and 21 squared is 441. These sum to give us 841. We can then take the square root of both sides of our equation. The square root of 841 is 29.
If the two rectangles 𝐴𝐵𝐶𝐷 and 𝑋𝐵𝑍𝑌 are similar, the length of the line segment 𝑋𝑍 is 29 centimeters.