Starting from the given proportional relationship:

\[
\frac{a+b}{b+c} = \frac{c+d}{d+a}
\]

Multiplying across to eliminate the denominators, we have:

\[
(a + b)(d + a) = (b + c)(c + d)
\]

Expanding both sides:

\[
ad + a^2 + bd + ab = bc + c^2 + cd + bd
\]

Rearranging to group like terms:

\[
a^2 – c^2 + ad – cd + ab – bc = 0
\]

Factoring by grouping, where appropriate, using the difference of squares for \(a^2 – c^2\) and factoring out the common terms in the other parts:

\[
(a – c)(a + c) + (a – c)d + (a – c)b = 0
\]

Factoring \(a – c\) from each term:

\[
(a – c)(a + c + d + b) = 0
\]

For this product to equal zero, at least one of the factors must be zero. Therefore:

\[
a – c = 0 \quad \text{or} \quad a + b + c + d = 0
\]

This means:

– \(a = c\), or
– \(a + b + c + d = 0\), or
– Both conditions could be true in certain scenarios.

Therefore, the correct interpretation is option (c) either \(a = c\) or \(a+b+c+d = 0\), or both.

Given the operation \(x^{*} y = x^{2} + y^{2} – xy\), let’s calculate the value of \(9^{*} 11\).

Substitute \(x = 9\) and \(y = 11\) into the formula:

\[
9^{*} 11 = 9^{2} + 11^{2} – 9 \times 11
\]

Simplify each term:

\[
9^{*} 11 = 81 + 121 – 99
\]

Add and subtract the terms:

\[
9^{*} 11 = 202 – 99 = 103
\]

Therefore, the value of \(9^{*} 11\) is \(\boldsymbol{103}\).

The expression \(|17x – 8| – 9\) represents the distance of \(17x – 8\) from 0 on the number line, minus 9. The minimum value of the absolute value function \(|17x – 8|\) is 0, which occurs when \(17x – 8 = 0\) or \(x = \frac{8}{17}\).

Since the absolute value function cannot be negative, the minimum value of \(|17x – 8|\) is 0. Therefore, the minimum value of the entire expression \(|17x – 8| – 9\) is \(0 – 9 = -9\).

Thus, the correct answer is (b) -9.

Let’s break this down step by step:

1. Volume of the Cube : The volume of a cube is given by \( \text{Volume} = a^3 \), where \( a \) is the side length of the cube. Here, \( a = 11 \) cm. So, the volume of the cube is \( 11^3 \) cubic cm.

2. Volume of the Cylinder : The volume of a cylinder is given by \( \text{Volume} = \pi r^2 h \), where \( r \) is the radius and \( h \) is the height. We are given that the height of the cylinder is 7 times the length of the rectangle. The length of the rectangle is not explicitly given, but we can find it using the given information about the width and perimeter of the rectangle.

3. Finding the Length of the Rectangle : The perimeter of a rectangle is given by \( 2(l + w) \), where \( l \) is the length and \( w \) is the width. We are given that the width \( w = 1.5 \) cm and the perimeter is 4 cm. Therefore, \( 2(l + 1.5) = 4 \). Solving for \( l \), we get \( l = \frac{4}{2} – 1.5 = 0.5 \) cm.

4. Height of the Cylinder : Since the height of the cylinder is 7 times the length of the rectangle, the height is \( 7 \times 0.5 = 3.5 \) cm.

5. Volume of the Cylinder (Continued) : Now that we have the height of the cylinder as 3.5 cm, we can find its volume using the formula \( \pi r^2 h \).

6. Equating Volumes : Since the cube is melted to form the cylinder, the volume of the cube should be equal to the volume of the cylinder. Setting these two volumes equal to each other, we can solve for the radius \( r \) of the cylinder.

Let’s calculate the radius \( r \) of the cylinder using the given information.

Given:
Side length of cube, \(a = 11\) cm
Width of rectangle, \(w = 1.5\) cm
Perimeter of rectangle, \(P = 4\) cm

1. Volume of Cube : \(V_{\text{cube}} = a^3 = 11^3\) cm³.

2. Length of Rectangle : Perimeter of rectangle, \(P = 2(l + w)\). We have \(P = 4\) and \(w = 1.5\). Solve for \(l\):

\[4 = 2(l + 1.5) \]
\[2 = l + 1.5 \]
\[l = 0.5\] cm.

3. Height of Cylinder : Height of cylinder, \(h = 7l = 7 \times 0.5\) cm.

4. Volume of Cylinder : Volume of cylinder, \(V_{\text{cylinder}} = \pi r^2 h\).

Since the cube is melted and converted into the cylinder, their volumes are equal:

\[11^3 = \pi r^2 \times 7 \times 0.5\]

\[1331 = 3.5 \pi r^2\]

\[r^2 = \frac{1331}{3.5\pi}\]

\[r = \sqrt{\frac{1331}{3.5\pi}}\]

To find the radius of the cylinder, we first calculate the value inside the square root:

\[ r = \sqrt{\frac{1331}{3.5\pi}} \]

\[ r = \sqrt{\frac{1331}{3.5 \times 3.14159}} \]

\[ r = \sqrt{\frac{1331}{10.99265}} \]

\[ r = \sqrt{121} \]

\[ r = 11 \text{ cm} \]

Therefore, the radius of the cylinder is 11 cm.

Solution

Given:
– The lengths of the sides of the triangle are 9 cm, 12 cm, and 15 cm.

We can use Heron’s formula to find the area of the triangle and then use the area to find the length of the perpendicular from the opposite vertex to the side of length 15 cm.

Step 1: Calculate the semi-perimeter (s) of the triangle

\[ s = \frac{9 + 12 + 15}{2} = 18 \text{ cm} \]

Step 2: Use Heron’s formula to find the area (A) of the triangle

\[ A = \sqrt{s(s – 9)(s – 12)(s – 15)} \]
\[ A = \sqrt{18(18 – 9)(18 – 12)(18 – 15)} \]
\[ A = \sqrt{18 \times 9 \times 6 \times 3} \]
\[ A = \sqrt{2916} \]
\[ A = 54 \text{ cm}^2 \]

Step 3: Find the length of the perpendicular (h) from the opposite vertex to the side of length 15 cm

Using the formula for the area of a triangle (\(A = \frac{1}{2} \times \text{base} \times \text{height}\)):
\[ 54 = \frac{1}{2} \times 15 \times h \]
\[ h = \frac{54 \times 2}{15} \]
\[ h = \frac{108}{15} \]
\[ h = 7.2 \text{ cm} \]

Conclusion

The length of the perpendicular from the opposite vertex to the side whose length is 15 cm is 7.2 cm.