For this case we have:
Question 1:
[tex]BC = 24\\AD = 5y-1[/tex]
By definition, one of the properties of the rectangle states that:
The opposite sides of a rectangle have the same length, that is, they are equal, then:
[tex]BC = AD[/tex]
So:
[tex]24 = 5y-1[/tex]
Clearing "y" we have:
[tex]24 + 1 = 5y\\25 = 5y\\y = \frac {25} {5}\\y = 5[/tex]
Thus, the value of "y" is 5.
Answer:
[tex]y = 5[/tex]
Question 2:
[tex]AC = 2x + 13\\DB = 4x-1[/tex]
By definition, one of the properties of the rectangles states that:
The diagonals of a rectangle have the same lengths, that is:
[tex]AC = DB\\2x + 13 = 4x-1[/tex]
We clear the value of "x":
[tex]13 + 1 = 4x-2x\\14 = 2x\\x = \frac {14} {2}\\x = 7[/tex]
We must find DB:
[tex]DB = 4x-1 \\DB = 4 (7) -1\\DB = 28-1\\DB = 27[/tex]
ANswer:
[tex]DB = 27[/tex]
Question 3:
[tex]AE = 3x + 3\\EC = 5x-15[/tex]
By definition, one of the properties of the rectangles states that:
The diagonals of a rectangle intersect and at the point of intersection they are divided in half, that is:
[tex]AE = EC\\3x + 3 = 5x-15[/tex]
Clearing the value of "x" we have:
[tex]3 + 15 = 5x-3x\\18 = 2x\\x = \frac {18} {2}\\x = 9[/tex]
We know that:
[tex]AC = AE + EC\\AC = 3x + 3 + 5x-15\\AC = 8x-12\\AC = 8 (9) -12\\AC = 60[/tex]
ANswer:
[tex]AC = 60[/tex]
Question 4:
[tex]m <ABD = 7x-31\\m <CDB = 4x + 5[/tex]
By definition, one of the properties of the rectangles states that:
The 4 angles of the rectangles are straight. From there it turns out that:
[tex]m <ABD = m <CBD\\7x-31 = 4x + 5[/tex]
We clear the value of "x":
[tex]7x-4x = 5 + 31\\3x = 36\\x = \frac {36} {3}\\x = 12[/tex]
So:
[tex]m <ABD = 7x-31\\m <ABD = 7 (12) -31\\m <ABD = 84-31\\m <ABD = 53[/tex]
Answer:
[tex]m <ABD = 53[/tex]
