Question

2 practice 2
in right triangle abc, a square is drawn on each of its sides. an altitude cd is drawn to the hypotenuse ab and extended to the opposite side of the square on fe. in class, we discussed elenas observation that ( a^{2}=x c ) and diegos observation that ( b^{2}=y c ). mai observes that these statements can be thought of as claims about the areas of rectangles.

a. which rectangle has the same area as bghc?
a rectangle ajic
b rectangle kebd
c rectangle fkda
d rectangle feba
b. which rectangle has the same area as aclj?
a rectangle chgb
b rectangle kebd
c rectangle fkda
d rectangle feba

Explanation:

Step1: Recall the area formula for rectangles

The area of a rectangle is \(A = \text{length}\times\text{width}\). For rectangle \(BGHC\), its area \(A_{BGHC}=a^{2}\) (since it's a square with side - length \(a\)). Given \(a^{2}=xc\), and for rectangle \(KEBD\), its length is \(c\) and width is \(x\), so \(A_{KEBD}=x\times c\).

Step2: Recall the area formula for rectangles (second part)

For rectangle \(ACIJ\), its area \(A_{ACIJ}=b^{2}\) (since it's a square with side - length \(b\)). Given \(b^{2}=yc\), and for rectangle \(FKDA\), its length is \(c\) and width is \(y\), so \(A_{FKDA}=y\times c\).

Answer:

a. B. Rectangle \(KEBD\)
b. C. Rectangle \(FKDA\)