# Geometry

## Educators ### Problem 1

If function $f : x \rightarrow 5 x-7,$ find the image of 8 and the preimage of $13 .$ Amrita B.

### Problem 2

If function $g : x \rightarrow 8-3 x,$ find the image of 5 and the preimage of $0 .$ Amrita B.

### Problem 3

If $f(x)=x^{2}+1,$ find $f(3)$ and $f(-3) .$ Is $f$ a one-to-one function? Amrita B.

### Problem 4

If $h(x)=6 x+1,$ find $h\left(\frac{1}{2}\right)$ . Is $h$ a one-to-one function? Amrita B.

### Problem 5

For each transformation given in Exercises $5-10 :$
$$T :(x, y) \rightarrow(x+4, y-2)$$ Amrita B.

### Problem 6

For each transformation given in Exercises $5-10 :$
$$S :(x, y) \rightarrow(2 x+4,2 y-2)$$ Amrita B.

### Problem 7

For each transformation given in Exercises $5-10 :$
$$D :(x, y) \rightarrow(3 x, 3 y)$$ Amrita B.

### Problem 8

For each transformation given in Exercises $5-10 :$
$$H :(x, y) \rightarrow(-x,-y)$$ Amrita B.

### Problem 9

For each transformation given in Exercises $5-10 :$
$$M :(x, y) \rightarrow(12-x, y)$$ Amrita B.

### Problem 10

For each transformation given in Exercises $5-10 :$
$$G :(x, y) \rightarrow\left(-\frac{1}{2} x,-\frac{1}{2} y\right)$$

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### Problem 11

$O$ is a point equidistant from parallel lines $l_{1}$ and $l_{2}$ . A mapping $M$ maps each point $P$ of $l_{1}$ to the point $P^{\prime}$ where $\overrightarrow{P O}$ intersects $l_{2}$ .
a. Is the mapping a one-to-one mapping of $l_{1}$ onto $l_{2} ?$
b. Does this mapping preserve or distort distance?
c. If $l_{1}$ and $l_{2}$ were not parallel, would the mapping preserve distance? Illustrate your answer with a sketch. Amrita B.

### Problem 12

$\triangle X Y Z$ is isosceles with $\overline{X Y} \cong \overline{X Z} .$ Describe a way of mapping
each point of $\overline{X Y}$ to a point of $\overline{X Z}$ so that the mapping is an isometry. Amrita B.

### Problem 13

$A B C D$ is a trapezoid. Describe a way of mapping each point of $\overline{D C}$ to a point of $\overline{A B}$ so that the mapping is one-to-one. Is your mapping an isometry? Amrita B.

### Problem 14

The red and blue squares are congruent and have the same center $O$ . A mapping maps each point $P$ of the red square to the point $P^{\prime}$ where $\overrightarrow{O P}$ intersects the blue square.
a. Is this mapping one-to-one?
b. Copy the diagram and locate a point $X$ that is its own image.
c. Locate two points $R$ and $S$ on the red square and their images $R^{\prime}$ and $S^{\prime}$ on the blue square that have the property that $R S \neq R^{\prime} S^{\prime}$ .
d. Does this mapping preserve distance?
e. Describe a mapping from the red square onto the blue square that does preserve distance.

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### Problem 15

The transformation $T :(x, y) \rightarrow(x+y, y)$ preserves areas of figures even though it does not preserve distances. Illustrate this by drawing a square with vertices $A(2,3), B(4,3), C(4,5),$ and $D(2.5)$ and its image $A^{\prime} B^{\prime} C^{\prime} D^{\prime}$ . Find the area and perimeter of each figure.

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### Problem 16

A piece of paper is wrapped around a globe of the Earth to form a cylinder as shown. $O$ is the center of the Earth and a point $P$ of the globe is projected along $O \vec{P}$ to a point $P^{\prime}$ of the cylinder.
Describe the image of the globe's equator. Amrita B.

### Problem 17

A piece of paper is wrapped around a globe of the Earth to form a cylinder as shown. $O$ is the center of the Earth and a point $P$ of the globe is projected along $O \vec{P}$ to a point $P^{\prime}$ of the cylinder.
Is the image of the Arctic Circle congruent to the image of the equator?

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### Problem 18

A piece of paper is wrapped around a globe of the Earth to form a cylinder as shown. $O$ is the center of the Earth and a point $P$ of the globe is projected along $O \vec{P}$ to a point $P^{\prime}$ of the cylinder.
Are distances near the equator distorted more than or less than distances near the Arctic Circle? Amrita B.

### Problem 19

A piece of paper is wrapped around a globe of the Earth to form a cylinder as shown. $O$ is the center of the Earth and a point $P$ of the globe is projected along $O \vec{P}$ to a point $P^{\prime}$ of the cylinder.
Does the North Pole (point $N$ ) have an image? Amrita B.

### Problem 20

A piece of paper is wrapped around a globe of the Earth to form a cylinder as shown. $O$ is the center of the Earth and a point $P$ of the globe is projected along $O \vec{P}$ to a point $P^{\prime}$ of the cylinder.
Consider the mapping $S :(x, y) \rightarrow(x, 0)$ .
a. Plot the points $P(4,5), Q(-3,2),$ and $R(-3,-1)$ and their images.
b. Does $S$ appear to be an isometry? Explain.
c. Is $S$ a transformation? Explain.

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### Problem 21

A piece of paper is wrapped around a globe of the Earth to form a cylinder as shown. $O$ is the center of the Earth and a point $P$ of the globe is projected along $O \vec{P}$ to a point $P^{\prime}$ of the cylinder.
Mapping $M$ maps points $A$ and $B$ to the same image point. Explain why the mapping $M$ does not preserve distance. Amrita B.

### Problem 22

A piece of paper is wrapped around a globe of the Earth to form a cylinder as shown. $O$ is the center of the Earth and a point $P$ of the globe is projected along $O \vec{P}$ to a point $P^{\prime}$ of the cylinder.
Fold a piece of paper. Cut a design connecting the top and bottom point of the fold, as shown. Unfold the shape. Consider a mapping $M$ of the points in the gray region to the corresponding points in the red region
a. Does $M$ appear to be an isometry?
b. If a point $P$ is on line $k,$ what is the image of $P ?$
c. If a point $Q$ is not on line $k,$ and $M(Q)=Q^{\prime},$ what is the relationship between line $k$ and $\frac{M(Q)}{Q Q^{\prime} ?}$

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### Problem 23

A piece of paper is wrapped around a globe of the Earth to form a cylinder as shown. $O$ is the center of the Earth and a point $P$ of the globe is projected along $O \vec{P}$ to a point $P^{\prime}$ of the cylinder.
a. Plot the points $A(6,1), B(3,4),$ and $C(1,-3)$ and their images $A^{\prime},$ $B^{\prime},$ and $C^{\prime}$ under the transformation $R :(x, y) \rightarrow(-x, y) .$
b. Prove that $R$ is an isometry. (Hint: Let $P\left(x_{1}, y_{1}\right)$ and $Q\left(x_{2}, y_{2}\right)$ be any two points. Find $P^{\prime}$ and $Q^{\prime},$ and use the distance formula to show that $P Q=P^{\prime} Q^{\prime} . )$

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