Question

find the value of a in the parallelogram.
(the parallelogram has sides labeled: ab: 2a - 19.9, bc: a - 3.5, cd: a + 1.1, ad: 17.5)
a = 21
(type an integer or a decimal.)
ab = 22.1
(type an integer or a decimal.)
bc = 17.5
(type an integer or a decimal.)
cd = 22.1
(type an integer or a decimal.)
ad = \square
(type an integer or a decimal.)

Explanation:

Step1: Recall parallelogram property

In a parallelogram, opposite sides are equal. So, \( BC = AD \) and \( AB = CD \). We are given \( AD = 17.5 \) (from the diagram) and we can also verify using the side \( BC=a - 3.5\). But for \( AD \), from the diagram, the length of \( AD \) is given as \( 17.5 \) (the arrowed side from \( A \) to \( D \) is labeled \( 17.5 \)). Also, we can check with the other pair of sides. For \( AB = 2a-19.9 \) and \( CD=a + 1.1 \), we know \( AB = CD \), but for \( AD \), since \( BC=a - 3.5 \) and \( BC = AD \), we can also calculate \( a \) first. Wait, but the diagram directly labels \( AD \) as \( 17.5 \)? Wait, no, the label on \( AD \) is \( 17.5 \), so \( AD = 17.5 \). Wait, but let's confirm with the side \( BC \). We know that in a parallelogram \( BC = AD \). We found \( a = 21 \) (from previous calculations, since \( AB=CD \) gives \( 2a-19.9=a + 1.1\), solving \( 2a - a=1.1 + 19.9\), \( a=21 \)). Then \( BC=a - 3.5=21 - 3.5 = 17.5 \), which matches \( AD \) (since \( AD = 17.5 \) as labeled). So \( AD = 17.5 \).

Step2: Confirm the length of \( AD \)

From the diagram, the side \( AD \) is labeled with length \( 17.5 \), and using the parallelogram property (opposite sides equal, \( BC = AD \)), and with \( a = 21 \), \( BC=21 - 3.5 = 17.5 \), so \( AD = 17.5 \).

Answer:

\( 17.5 \)