Question

juanita is cutting a piece of construction paper in the shape of a parallelogram. two opposite sides of the parallelogram have lengths (5n - 6) cm and (3n - 2) cm. a third side measures (2n + 3) cm. what are the lengths of two adjacent sides of the parallelogram? 2 cm and 2 cm 4 cm and 7 cm 7 cm and 9 cm 13 cm and 19 cm

Explanation:

Step1: Recall parallelogram property

In a parallelogram, opposite sides are equal. So we have two pairs of equal sides. Let's assume the sides are \(5n - 6\), \(3n - 2\), and \(2n + 3\). Since opposite sides are equal, we can set \(5n - 6 = 2n + 3\) (assuming these are opposite sides) or \(3n - 2 = 2n + 3\). Let's try \(3n - 2 = 2n + 3\) first.

Step2: Solve for n (first equation)

\(3n - 2 = 2n + 3\)
Subtract \(2n\) from both sides: \(3n - 2n - 2 = 2n - 2n + 3\)
Simplify: \(n - 2 = 3\)
Add 2 to both sides: \(n = 3 + 2 = 5\). Wait, but let's check with the other equation. Now try \(5n - 6 = 2n + 3\)

Step3: Solve for n (second equation)

\(5n - 6 = 2n + 3\)
Subtract \(2n\) from both sides: \(5n - 2n - 6 = 2n - 2n + 3\)
Simplify: \(3n - 6 = 3\)
Add 6 to both sides: \(3n = 3 + 6 = 9\)
Divide by 3: \(n = \frac{9}{3} = 3\)

Step4: Calculate side lengths

Now, substitute \(n = 3\) into the expressions:
- \(5n - 6 = 5(3) - 6 = 15 - 6 = 9\)
- \(3n - 2 = 3(3) - 2 = 9 - 2 = 7\)
- \(2n + 3 = 2(3) + 3 = 6 + 3 = 9\) (Wait, no, maybe I mixed up. Wait, the problem says two opposite sides are \(5n - 6\) and \(3n - 2\) is another, and \(2n + 3\) is the third. Wait, no, in a parallelogram, there are two pairs of equal sides. So let's correct. Let's say the sides are \(5n - 6\), \(3n - 2\), \(5n - 6\), \(3n - 2\) (since opposite sides are equal). But the problem mentions a third side \(2n + 3\), which must be equal to one of the other sides. So either \(5n - 6 = 2n + 3\) or \(3n - 2 = 2n + 3\). We saw \(n = 3\) from \(5n - 6 = 2n + 3\) gives \(5(3)-6 = 9\), \(2(3)+3 = 9\), so those are equal. Then the other side is \(3n - 2 = 3(3)-2 = 7\). So the sides are 9, 7, 9, 7? Wait, but the options are 2 and 2, 4 and 7, 7 and 9, 13 and 19. Wait, maybe I made a mistake. Let's try \(3n - 2 = 2n + 3\) gives \(n = 5\). Then \(5n - 6 = 5(5)-6 = 19\), \(3n - 2 = 13\), \(2n + 3 = 13\). Then sides are 19,13,19,13. But that's an option (13 and 19). Wait, but the problem says "two opposite sides of the parallelogram have lengths \((5n - 6)\) cm and \((3n - 2)\) cm. A third side measures \((2n + 3)\) cm." Wait, no, in a parallelogram, there are four sides: two pairs of equal sides. So if two opposite sides are \(5n - 6\) and \(3n - 2\), that can't be, because opposite sides must be equal. So actually, the two opposite sides are \(5n - 6\) and \(2n + 3\) (since they are equal), and the other two opposite sides are \(3n - 2\) and \(2n + 3\)? No, that's not right. Wait, correct property: In a parallelogram, opposite sides are congruent. So we have two pairs: let's say side 1 = side 3, side 2 = side 4. So if the sides are \(5n - 6\), \(3n - 2\), \(5n - 6\), \(3n - 2\), but there's a third side \(2n + 3\), which must be equal to either \(5n - 6\) or \(3n - 2\). So case 1: \(5n - 6 = 2n + 3\)
\(5n - 2n = 3 + 6\)
\(3n = 9\)
\(n = 3\)
Then \(5n - 6 = 9\), \(3n - 2 = 7\), \(2n + 3 = 9\). So the sides are 9,7,9,7. So adjacent sides are 7 and 9. Case 2: \(3n - 2 = 2n + 3\)
\(n = 5\)
Then \(5n - 6 = 19\), \(3n - 2 = 13\), \(2n + 3 = 13\). So sides are 19,13,19,13. Adjacent sides 13 and 19. Now check the options. The options are 2 and 2, 4 and 7, 7 and 9, 13 and 19. So both 7 and 9 (when n=3) and 13 and 19 (when n=5) are options. Wait, but let's check if the side lengths are positive. For n=3: \(5n -6 = 9\) (positive), \(3n -2 =7\) (positive), \(2n +3 =9\) (positive). For n=5: \(5n -6 =19\), \(3n -2 =13\), \(2n +3 =13\) (all positive). But the problem says "cutting a piece of construction paper", so maybe smaller lengths. Wait, maybe I misread the problem. Wait, the problem says "two opposite sides of the parallelogram have lengths \((5n - 6)\) cm and \((3n - 2)\) cm. A third side measures \((2n + 3)\) cm." Wait, no, in a parallelogram, there are four sides: two pairs. So if two opposite sides are \(5n -6\) and \(3n -2\), that's a contradiction because opposite sides must be equal. So the correct interpretation is: The parallelogram has sides \(5n -6\), \(3n -2\), \(5n -6\), \(3n -2\), and a third side (wait, no, four sides). Wait, maybe the problem means that among the four sides, two opposite sides are \(5n -6\) and \(3n -2\) (which is wrong, so actually, the two opposite sides are \(5n -6\) and \(2n +3\), and the other two are \(3n -2\) and \(2n +3\). So when \(5n -6 = 2n +3\), n=3, sides 9,7,9,7 (adjacent 7 and 9). When \(3n -2 = 2n +3\), n=5, sides 19,13,19,13 (adjacent 13 and 19). Now check the options: 7 cm and 9 cm is an option, 13 and 19 is also. Wait, maybe I made a mistake in the equation. Let's re-express:

In a parallelogram, opposite sides are equal. So let's denote the sides as \(a\), \(b\), \(a\), \(b\). The problem says two opposite sides are \(5n -6\) and \(3n -2\) – that can't be, so actually, the two opposite sides are \(5n -6\) and \(2n +3\) (so \(5n -6 = 2n +3\)), and the other two opposite sides are \(3n -2\) and \(2n +3\) (so \(3n -2 = 2n +3\))? No, that would mean \(5n -6 = 2n +3\) and \(3n -2 = 2n +3\), which would require \(n=3\) and \(n=5\), impossible. So the correct way is: The four sides are \(5n -6\), \(3n -2\), \(5n -6\), \(3n -2\), and one of these is equal to \(2n +3\). So either \(5n -6 = 2n +3\) or \(3n -2 = 2n +3\).

First equation: \(5n -6 = 2n +3\)
\(3n = 9\)
\(n=3\)
Then \(5n -6 = 9\), \(3n -2 =7\), \(2n +3 =9\). So sides are 9,7,9,7. Adjacent sides 7 and 9.

Second equation: \(3n -2 = 2n +3\)
\(n=5\)
Then \(5n -6 =19\), \(3n -2 =13\), \(2n +3 =13\). Sides 19,13,19,13. Adjacent sides 13 and 19.

Now check the options: 7 cm and 9 cm is option C, 13 and 19 is option D. Wait, but let's check if the side lengths make sense for a parallelogram (sum of two sides > third side, triangle inequality, but in parallelogram, it's a quadrilateral, so sum of adjacent sides > opposite side? No, the triangle inequality for the parallelogram: when you split it into two triangles, the sum of two sides must be greater than the diagonal, but for the sides themselves, adjacent sides can be any positive lengths as long as they are positive. For n=3: sides 7 and 9, both positive. For n=5: 13 and 19, positive. But the problem says "cutting a piece of construction paper", so likely smaller lengths. So 7 and 9 cm. Wait, but let's check the problem statement again: "two opposite sides of the parallelogram have lengths \((5n - 6)\) cm and \((3n - 2)\) cm. A third side measures \((2n + 3)\) cm." Wait, that means there are three sides mentioned: two opposite sides (which should be equal, so \(5n -6 = 3n -2\)), and a third side. Wait, that's the key! I missed that. The problem says "two opposite sides of the parallelogram have lengths \((5n - 6)\) cm and \((3n - 2)\) cm". But in a parallelogram, opposite sides are equal, so \(5n -6 = 3n -2\). That's the correct equation! Oh! I made a mistake earlier. So:

Step5: Correct equation (opposite sides equal)

Since two opposite sides are \(5n -6\) and \(3n -2\), they must be equal. So:
\(5n - 6 = 3n - 2\)

Step6: Solve for n

Subtract \(3n\) from both sides: \(5n - 3n - 6 = 3n - 3n - 2\)
Simplify: \(2n - 6 = -2\)
Add 6 to both sides: \(2n = -2 + 6 = 4\)
Divide by 2: \(n = \frac{4}{2} = 2\)

Step7: Calculate side lengths

Now, substitute \(n = 2\) into the expressions:
- \(5n - 6 = 5(2) - 6 = 10 - 6 = 4\)
- \(3n - 2 = 3(2) - 2 = 6 - 2 = 4\)? No, that can't be. Wait, no, the third side is \(2n + 3\). Wait, no, the problem says "two opposite sides... have lengths \((5n - 6)\) and \((3n - 2)\)". A third side is \((2n + 3)\). So in a parallelogram, there are four sides: two pairs. So if two opposite sides are \(5n -6\) and \(3n -2\) (which must be equal), so \(5n -6 = 3n -2\) gives \(n=2\), then those sides are 4 cm. Then the third side is \(2n +3 = 7\) cm. So the sides are 4,7,4,7. So adjacent sides are 4 and 7. Wait, that's option B (4 cm and 7 cm). Oh! I see my mistake earlier. I thought the third side was equal to one of the other two, but the problem says "two opposite sides" (which must be equal) are \(5n -6\) and \(3n -2\), so set them equal. Then the third side is \(2n +3\), which is equal to the other pair of opposite sides. So:

\(5n -6 = 3n -2\) (opposite sides)
\(2n = 4\)
\(n=2\)
Then \(5n -6 = 4\), \(3n -2 =4\), \(2n +3 =7\). So the sides are 4,7,4,7. So adjacent sides are 4 and 7. That's option B. Wait, but let's verify:

If \(n=2\), then \(5n -6 = 10 -6 =4\), \(3n -2 =6 -2 =4\), \(2n +3 =4 +3 =7\). So the parallelogram has sides 4,7,4,7. So adjacent sides are 4 and 7. That makes sense. I had misinterpreted which sides are opposite. The problem says "two opposite sides of the parallelogram have lengths \((5n - 6)\) cm and \((3n - 2)\) cm", so those two must be equal (since opposite sides in parallelogram are equal). Then the third side is \(2n +3\), which is equal to the other pair of opposite sides. So that's the correct approach.

Answer:

4 cm and 7 cm (Option B)