The area of a rectangle is 45 cm². two squares are constructed such that two adjacent sides of the rectangle are each also the side of one of the squares. the combined area of the two squares is 106 cm². find the lengths of the sides of the squares.

5 cm, 9 cm

Step-by-step explanation:

If x and y are the lengths of sides of the rectangle, then the problem statement tells you ...

Substituting y = 45/x into the second equation, you get

... x² + (45/x)² = 106

Multiplying by x² gives ...

... x⁴ -106x² +2025 = 0 . . . . . after subtracting 106x²

... (x² -53)² -784 = 0 . . . . . . . . complete the square

... x² = 53 ±√784 = 53 ±28 . . . solve for x²

... x = √81 or √25 = 9 or 5 . . . . solve for x (which must be positive)

y is the other of these two values, so ...

The lengths of the sides of the squares are 5 cm and 9 cm.

_____

Comment on the graph

The graph is symmetrical about the origin, so there are also two solutions at (x, y) = (-5, -9) and (-9, -5). Since x and y represent distances, these are extraneous solutions with respect to the original question.

δabc ≅ δdcb by sas because

db ≅ ac . . given (side)

bc ≅ bc . . reflexive property (side)

∠dbc ≅ ∠ acb . . alternate interior angles where transversal bc crosses parallel lines bd and ac (included angle)

80° is the measure of gh

step-by-step explanation:

circle f is congruent to circle j

we have two circle with center f and j.

both circles are congruent. the radius of the circle would be equal.

see the attachment for the circle.

we are given ∠efd ≅ ∠gjh

therefore, arc (ed) = arc (gh)     (∴ circle f is congruent to circle j)

but ∠dfe=80°=arc (ed)

therefore, arc (gh) = 80°

thus, 80° is the measure of gh