# Lesson 4

Scaling and Area

- Let’s see how the area of shapes changes when we dilate them.

### Problem 1

A rectangle with area 12 square units is dilated by a scale factor of \(k\). Find the area of the image for each given value of \(k\).

- \(k=2\)
- \(k=5\)
- \(k=1\)
- \(k=\frac14\)
- \(k=1.2\)

### Problem 2

The area of a circle of radius 1 is \(\pi\) units squared. Use scaling to explain why the area of a circle of radius \(r\) is \(\pi r^2\) units squared.

### Problem 3

Trapezoid \(A’B’C’D\) was created by dilating trapezoid \(ABCD\) using \(D\) as the center of dilation.

- What was the scale factor of the dilation?
- Based on the scale factor, how many copies of \(ABCD\), including the original, should fit inside \(A’B’C’D\)?
- How can you see your answer to these questions in the diagram?

### Problem 4

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. Estimate the scale factor that was used for each dilation.

### Problem 5

Select the solid whose cross sections are dilations of some two-dimensional shape using a point directly above the shape as a center and scale factors ranging from 0 to 1.

cone

cube

cylinder

triangular prism

### Problem 6

Select **all** figures for which at least one cross section is a circle.

triangular pyramid

square pyramid

rectangular prism

cube

cone

cylinder

sphere

### Problem 7

If the two-dimensional figures are rotated around the vertical axes of rotation shown, what solids are formed?

### Problem 8

Tyler and Jada wish to find the value of \(x\), the length of side \(BC\) in this triangle. Tyler decides to set up the equation \(\tan(56)=\frac8x\). Jada says she prefers an equation that has \(x\) in the numerator. What is an equation she could use instead?

### Problem 9

Triangles \(ACD\) and \(BCD\) are isosceles. Angle \(DBC\) has a measure of 110 degrees and angle \(BDA\) has a measure of 22 degrees. Find the measure of angle \(BAC\).