Question

question 4 points 3
find the measure of the side bc in the parallelogram abcd.
parallelogram abcd with side bc labeled 2x - 24 and side ad labeled x + 6
options: 36, 30, 18, 15

Explanation:

Step1: Recall parallelogram property

In a parallelogram, opposite sides are equal. So, \( BC = AD \).
Given \( BC = 2x - 24 \) and \( AD = x + 6 \), we set up the equation: \( 2x - 24 = x + 6 \).

Step2: Solve for \( x \)

Subtract \( x \) from both sides: \( 2x - x - 24 = x - x + 6 \), which simplifies to \( x - 24 = 6 \).
Add 24 to both sides: \( x - 24 + 24 = 6 + 24 \), so \( x = 30 \).

Step3: Find length of \( BC \)

Substitute \( x = 30 \) into \( BC = 2x - 24 \): \( BC = 2(30) - 24 = 60 - 24 = 36 \)? Wait, no, wait. Wait, let's check again. Wait, \( AD = x + 6 \), if \( x = 30 \), \( AD = 30 + 6 = 36 \), and \( BC = 2x - 24 = 60 - 24 = 36 \). But wait, the options have 36? Wait, no, wait, maybe I made a mistake. Wait, no, let's re - check the equation. Wait, in a parallelogram, \( BC \) and \( AD \) are opposite sides, so \( BC = AD \). So \( 2x - 24 = x + 6 \). Solving: \( 2x - x = 6 + 24 \), \( x = 30 \). Then \( BC = 2(30)-24 = 36 \), but wait, the option 36 is there. Wait, but let me check again. Wait, maybe the problem was misread. Wait, no, the diagram shows \( BC = 2x - 24 \) and \( AD = x + 6 \). So opposite sides equal, so equation is correct. So \( x = 30 \), \( BC = 2*30 - 24 = 36 \). But wait, the options include 36. Wait, but let me check the calculation again. \( 2x - 24 = x + 6 \), subtract \( x \): \( x - 24 = 6 \), add 24: \( x = 30 \). Then \( BC = 2*30 - 24 = 60 - 24 = 36 \). So the length of \( BC \) is 36? But wait, the options have 36 as the first option. Wait, but maybe I made a mistake. Wait, no, let's check the arithmetic. 2*30 is 60, 60 - 24 is 36. And \( AD = 30 + 6 = 36 \), so that's correct.

Wait, but the user's options: 36, 30, 18, 15. So 36 is an option. So the answer should be 36? Wait, but let me check again. Wait, maybe the problem was that I misassigned the sides. Wait, in parallelogram \( ABCD \), \( AB \) is adjacent to \( BC \), and \( AD \) is opposite to \( BC \). So yes, \( BC \) and \( AD \) are opposite, so they should be equal. So the calculation seems correct.

Wait, but wait, maybe I made a mistake in the equation. Let's re - derive the equation. In a parallelogram, opposite sides are congruent. So \( BC \cong AD \), so \( BC = AD \). Given \( BC = 2x - 24 \) and \( AD = x + 6 \), so \( 2x - 24 = x + 6 \). Solving for \( x \):

\( 2x - x = 6 + 24 \)

\( x = 30 \)

Then \( BC = 2(30)-24 = 60 - 24 = 36 \). So the length of \( BC \) is 36.

Answer:

36