# Lesson 3

Creating Cross Sections by Dilating

- Let’s create cross sections by doing dilations.

### Problem 1

Each image shows a quadrilateral in a plane. The quadrilateral has been dilated using a center above the plane and a scale factor between 0 and 1. Match the dilation with the scale factor used.

### Problem 2

The Pyramid of Khufu in Giza, Egypt was the world’s tallest free-standing structure for more than 3,500 years. Its original height was about 144 meters. Its base is approximately a square with a side length of 231 meters.

The diagram shows a cross section created by dilating the base using the top of the pyramid as the center of dilation. The cross section is at a height of 96 meters.

- What scale factor was used to create the cross section?
- What are the dimensions of the cross section?

### Problem 3

The horizontal cross sections of this figure are dilations of the bottom rectangle using a point above the rectangle as a center. What scale factors of dilation are represented in the figure’s cross sections?

scale factors between \(0\) and \(\frac{1}{2}\)

scale factors between \(0\) and \(1\)

scale factors between \(\frac{1}{4}\) and \(\frac{3}{4}\)

scale factors between \(\frac{1}{2}\) and \(1\)

### Problem 4

Imagine an upright cone with its base resting on your horizontal desk. Match each plane with the image of the cross section formed by intersecting the plane with the cone.

### Problem 5

What is the shape of the cross section formed by intersecting a cube with a vertical plane that passes through opposite edges of the cube? Explain how you know.

### Problem 6

Sketch the solid of rotation formed by rotating the given two-dimensional figure using the dashed vertical line as an axis of rotation.

### Problem 7

*Technology required. *A rope with a length of 4 meters is tied from a stake in the ground to the top of a tent. It forms a 20 degree angle with the ground. How tall is the tent?

### Problem 8

*Technology required. *What is the value of \(y\)?