### Video Transcript

Given that the rectangle π΄π΅πΆπ· is similar to the rectangle ππ΅ππ, find the length of line segment ππ.

We are told in the question that the rectangles π΄π΅πΆπ· and ππ΅ππ are similar. This means that theyβre enlargements or dilations of each other. The corresponding lengths in each rectangle will, therefore, be in the same ratio. The ratio of side lengths ππ and π΄π· will be equal to the ratio of side lengths ππ and πΆπ·. Substituting the lengths in centimeters from our diagram, we have ππ 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 ππ is equal to 20 centimeters.

We can now use the smaller rectangle to find the length of ππ. The Pythagorean theorem states that, in any right triangle, π squared plus π squared is equal to π squared, where π and π are the lengths of the shorter sides of the right triangle and π is the length of the hypotenuse. In the right triangle πππ, ππ is the hypotenuse. Therefore, ππ squared plus ππ squared is equal to ππ squared. Substituting in the lengths of ππ and ππ, 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 π΄π΅πΆπ· and ππ΅ππ are similar, the length of the line segment ππ is 29 centimeters.