Question

find the length
determine the length of the line segment(s) from each diagram.
1.
2.
determine the sum of the line segments for each diagram below.
3.
4.

Explanation:

1. For the first diagram in the "Find the Length" section:

• Assume the rectangles are adjacent and the widths of the two rectangles are the same. Let the width be \(w\). If the areas of the two rectangles are \(A_1 = 27\) units² and \(A_2=12\) units². Let the lengths of the two - rectangles be \(l_1\) and \(l_2\) respectively. We know that \(A_1=l_1w\) and \(A_2 = l_2w\). Then \(l_1=\frac{27}{w}\) and \(l_2=\frac{12}{w}\). The length of the line - segment \(AB=l_1 + l_2=\frac{27 + 12}{w}=\frac{39}{w}\). Without knowing the width \(w\), if we assume the rectangles have integer - valued side lengths and we find the greatest common divisor of 27 and 12. The factors of 27 are 1, 3, 9, 27 and the factors of 12 are 1, 2, 3, 4, 6, 12. The greatest common divisor of 27 and 12 is 3. If \(w = 3\), then \(l_1 = 9\) and \(l_2 = 4\), and the length of \(AB=9 + 4=13\) units.

1. For the second diagram in the "Find the Length" section:

• Let the widths of the three rectangles be the same, say \(w\). The areas of the rectangles are \(A_1 = 24\) units², \(A_2=96\) units², and \(A_3 = 27\) units². Let the lengths be \(l_1\), \(l_2\), and \(l_3\) respectively. Then \(l_1=\frac{24}{w}\), \(l_2=\frac{96}{w}\), \(l_3=\frac{27}{w}\). The length of the line - segment is \(l_1 + l_2+l_3=\frac{24 + 96+27}{w}=\frac{147}{w}\). If we assume \(w = 3\) (by looking at the common factors of 24, 96, and 27. The greatest common divisor of 24, 96, and 27 is 3). Then \(l_1 = 8\), \(l_2 = 32\), \(l_3 = 9\), and the length of the line - segment is \(8 + 32+9=49\) units.

1. For the first diagram in the "Determine the sum of the line segments" section:

• Let the width of the rectangles be \(w\). For the rectangle with area \(A_1 = 27\) units² and the rectangle with area \(A_2 = 12\) units². Let the lengths of the non - overlapping parts of the rectangles (the line - segments \(AB\) and \(CD\)) be considered. If we assume \(w = 3\) (since \(\gcd(27,12)=3\)). For the 27 - unit² rectangle, if \(w = 3\), the length of the horizontal part related to it is 9. For the 12 - unit² rectangle, if \(w = 3\), the length of the horizontal part related to it is 4. The sum of the lengths of \(AB\) and \(CD\): assume the rectangles are arranged such that we can find the lengths of the line - segments. If the rectangles are arranged in a way that we can separate the lengths, and with \(w = 3\), the sum of the lengths of \(AB\) and \(CD\) is \(9+4 = 13\) units.

1. For the second diagram in the "Determine the sum of the line segments" section:

• Let the width of the rectangles be \(w\). For the large rectangle with area \(A_1 = 96\) units² and the small rectangle with area \(A_2 = 24\) units². The greatest common divisor of 96 and 24 is 24. If \(w = 24\), for the 96 - unit² rectangle, the length is 4, and for the 24 - unit² rectangle, the length is 1. The sum of the lengths of \(AB\) and \(CD\): assume the rectangles are arranged in a way that we can measure the non - overlapping lengths. If we assume the rectangles are arranged appropriately, and with \(w = 24\), the sum of the lengths of \(AB\) and \(CD\) is \(4+(4 - 1)=7\) units.

Step1: Analyze the area - length relationship

For each rectangle, use the formula \(A=lw\) (where \(A\) is the area, \(l\) is the length, and \(w\) is the width). Given the areas of rectangles, express the lengths in terms of the width.

Step2: Find common factors

Find the greatest common divisor of the areas of the rectangles to assume a possible value for the width \(w\). This helps in getting integer - valued lengths for the rectangles.

Step3: Calculate the lengths of line - segments

Based on the assumed width and the area formula, calculate the lengths of the required line - segments.

Answer:

1. 13 units (assuming width \(w = 3\))
2. 49 units (assuming width \(w = 3\))
3. 13 units (assuming width \(w = 3\))
4. 7 units (assuming width \(w = 24\))