referring to the figure, the top of a right pyramid with sqaure base is cut and a frustmun ABCDRSPQ is formed. the upper side and the lower sides are 14cm and 24cm respecitvly. QC = 13cm. Find XY

This problem involves finding the distance \( XY \) in a frustum of a right pyramid with a square base. Let's break it down step-by-step to solve the problem.

### Step 1: Understanding the Geometry

- We have a frustum of a right pyramid, which means it's the section of the pyramid formed by cutting off the top part parallel to the base.
- The base of the frustum is square, with the length of the sides:
  - Upper base side length: 14 cm
  - Lower base side length: 24 cm
- The distance between points \( Q \) and \( C \) is given as 13 cm.

### Step 2: Identifying the Important Points and Planes
In a frustum, the points on the upper and lower square bases are aligned such that the top and bottom faces are similar and parallel. \( X \) and \( Y \) are likely referring to points that lie on the same plane, so we can calculate \( XY \) using similar triangles or geometrical properties.

### Step 3: Similar Triangles
Since the frustum is formed by a right pyramid, and the top and bottom faces are similar, the sides and diagonals of the upper and lower bases are proportional. The key is to use the information that the sides of the top and bottom squares are 14 cm and 24 cm respectively.

The diagonals of each square can be calculated using the Pythagorean theorem. For a square with side length \( s \), the diagonal \( d \) is given by:
\[
d = s\sqrt{2}
\]
Thus:
- Diagonal of the upper square = \( 14\sqrt{2} \)
- Diagonal of the lower square = \( 24\sqrt{2} \)

Since \( QC = 13 \) cm, we can calculate \( XY \) using the ratio of the corresponding sides of the upper and lower squares.

### Step 4: Using Proportions
The ratio of the sides of the frustum's upper and lower bases is:
\[
\text{Ratio} = \frac{14}{24} = \frac{7}{12}
\]
If \( XY \) lies on a line connecting corresponding points on the two squares, and given that \( QC = 13 \), we can apply the same ratio to find \( XY \):
\[
XY = QC \times \frac{7}{12}
\]
\[
XY = 13 \times \frac{7}{12} = \frac{91}{12} \approx 7.58 \text{ cm}
\]

Thus, the length of \( XY \) is approximately **7.58 cm**.

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Would you like further clarification on any step, or a more detailed explanation?

### Related Questions:
1. What is the total volume of the frustum given these dimensions?
2. How would you calculate the slant height of the frustum?
3. Can you determine the surface area of the frustum, including the lateral surface?
4. What is the height of the original pyramid before it was cut?
5. How would you find the height of the frustum if not given?

### Tip:
When solving problems involving similar figures, proportions are often a helpful tool to simplify complex geometry.

Referring to the figure, the top of a right pyramid with square base is cut and a frustum ABCDRSPQ is formed. The upper side and the lower sides are 14 cm and 24 cm respectively. QC = 13 cm. Find XY.

Learn how to find the length XY in a frustum formed by cutting a right pyramid with a square base. This problem involves geometry concepts like similar triangles, proportions, and the Pythagorean theorem. Suitable for students in Grades 9-12.

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