Question

find the value of a in the parallelogram.

a b c d parallelogram, with ab labeled 2a - 19.9, bc labeled a - 3.5, cd labeled a + 1.1, ad labeled 17.5.

a =
(type an integer or a decimal.)

Explanation:

Step1: Recall parallelogram side property

In a parallelogram, opposite sides are equal. So, \( BC = AD \) and \( AB = CD \). Let's use \( BC = AD \) first. \( BC = a - 3.5 \) and \( AD = 17.5 \), but also \( AB = 2a - 19.9 \) and \( CD = a + 1.1 \), so we can use \( AB = CD \) or \( BC = AD \). Let's use \( BC = AD \): \( a - 3.5 = 17.5 \)? Wait, no, maybe \( AB = CD \). Wait, \( AB = 2a - 19.9 \), \( CD = a + 1.1 \), so \( 2a - 19.9 = a + 1.1 \). Wait, or \( BC = AD \): \( a - 3.5 = 17.5 \)? Wait, let's check the diagram. \( AD \) is 17.5, \( BC \) is \( a - 3.5 \), so \( a - 3.5 = 17.5 \)? Wait, no, maybe I mixed up. Wait, in parallelogram, \( AB \) and \( CD \) are opposite, \( BC \) and \( AD \) are opposite. So \( AB = CD \): \( 2a - 19.9 = a + 1.1 \). Or \( BC = AD \): \( a - 3.5 = 17.5 \). Let's solve \( a - 3.5 = 17.5 \): \( a = 17.5 + 3.5 = 21 \). Wait, but let's check with \( AB = CD \). If \( a = 21 \), then \( AB = 2*21 - 19.9 = 42 - 19.9 = 22.1 \), \( CD = 21 + 1.1 = 22.1 \), which matches. And \( BC = 21 - 3.5 = 17.5 \), which matches \( AD = 17.5 \). So that works. Wait, so first, use \( BC = AD \): \( a - 3.5 = 17.5 \).

Step2: Solve for \( a \)

From \( a - 3.5 = 17.5 \), add 3.5 to both sides: \( a = 17.5 + 3.5 = 21 \). Let's verify with \( AB = CD \): \( AB = 2*21 - 19.9 = 42 - 19.9 = 22.1 \), \( CD = 21 + 1.1 = 22.1 \), which is equal. So \( a = 21 \).

Answer:

\( 21 \)