Question

an artist joined three square regions at their vertices to create the figure shown in the diagram. the artist will use small congruent square tiles to cover each region without any gaps or overlaps. based on the information, which statement is true? a the number of tiles needed to cover region z is the same as the number of tiles needed to cover both region x and region y. b none of these. c the number of tiles needed to cover region x is the same as the number of tiles needed to cover both region y and region z. d the number of tiles needed to cover region y is the same as the number of tiles needed to cover both region x and region z.

Explanation:

Step1: Recall Pythagorean theorem

For right - angled triangle formed by sides of squares X, Y, Z, if side length of square X is \(a\), side length of square Y is \(b\), and side length of square Z is \(c\), then \(a^{2}+b^{2}=c^{2}\) (Pythagorean theorem).

Step2: Relate to area

The area of square X is \(A_X=a^{2}\), the area of square Y is \(A_Y = b^{2}\), and the area of square Z is \(A_Z=c^{2}\). So \(A_X + A_Y=A_Z\).

Step3: Consider number of tiles

Since the number of congruent square tiles needed to cover a region is proportional to the area of the region, the number of tiles needed to cover Region Z is the same as the number of tiles needed to cover both Region X and Region Y.

Answer:

A. The number of tiles needed to cover Region Z is the same as the number of tiles needed to cover both Region X and Region Y