Question

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.

dilation a dilation b dilation c

1. $\frac{1}{4}$ 2. $\frac{1}{2}$ 3. $\frac{3}{4}$

a. dilation a b. dilation b c. dilation c

1. the pyramid of khufu in giza, egypt was the worlds 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.
a. what scale factor was used to create the cross section?
b. what are the dimensions of the cross section?

1. 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 figures cross sections?

a. scale factors between 0 and $\frac{1}{2}$
b. scale factors between 0 and 1
c. scale factors between $\frac{1}{4}$ and $\frac{3}{4}$
d. scale factors between $\frac{1}{2}$ and 1

Explanation:

Problem 1

Step1: Relate size to scale factor

Smaller image = smaller scale factor.

Step2: Match dilations to factors

Dilation B (smallest) → $\frac{1}{4}$; Dilation C → $\frac{1}{2}$; Dilation A (largest) → $\frac{3}{4}$.

Part a:

Step1: Use height ratio for scale

Scale factor = $\frac{\text{Cross section height}}{\text{Total pyramid height}}$
$\text{Scale factor} = \frac{96}{144}$

Step2: Simplify the fraction

$\frac{96}{144} = \frac{2}{3}$

Part b:

Step1: Scale base side length

Cross section side = $\text{Base length} \times \text{Scale factor}$
$\text{Cross section side} = 231 \times \frac{2}{3}$

Step2: Calculate the dimension

$231 \times \frac{2}{3} = 154$

The cross sections are dilations of the bottom rectangle (scale factor = 1 at the base) and shrink as they move up toward the top (scale factor > $\frac{1}{2}$, since the top is narrower than the base but not less than half its width).

Answer:

1. A Dilation A: $\boldsymbol{\frac{3}{4}}$
2. B Dilation B: $\boldsymbol{\frac{1}{4}}$
3. C Dilation C: $\boldsymbol{\frac{1}{2}}$

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Problem 2