If DE = 4x + 1, EB = 12x – 31, and CD = 28, find AD.

The value of AD in rectangle ABCD is 19.30 units

The figure above is a rectangle.

Therefore,

DE = EB

4x + 1 = 12x - 31

4x - 12x = -31 - 1

-8x = -32

x = -32 / -8

x = 4

CD = 28

DB = 2(DE)

DB = 2(4(4) + 1) = 34

DB = AC

AD² = 34² - 28²

AD² = 1156 - 784

AD = √372

AD = 19.287301522

AD ≈ 19.30 units

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AD = √372 = 19.3 units  

Step-by-step explanation:

If we find the length of the diagonal, using the property of the rectangles, that is diagonals bisect each other.

So DE = EB  

4x+ 1 = 12x - 31

Grouping the terms, we will get

12x - 4x = 31+ 1

8x = 32  

x= 32/8 = 4  

Now we can find the length of the diagonal as,  

DE + EB = (4(4) +1) + (12(4) - 31)  

= (16+1) + (48-31)

= 17+ 17 = 34 units.

Now we can find the height.  

Diagonal is equal to the sqrt of the sum of squares of width and height.

34 = √(CD²+ AD²)

Squaring on both sides, we will get,  

34² = (√(CD²+ AD²))²

1156 = CD²+ AD²

1156 = 28² + AD²

1156 = 784 + AD²

1156 - 784 = AD²

372 = AD²

AD = √372 = 19.3 units  

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