Math / Geometry

Question3. Reflect $\triangle S T U$ across line $S T$ . Which of these is a valid reason why the image of $U$ will coincide with $J$ ? a. The image of $U$ and $J$ are on the same side of $S T$ and make the same angle with it at $T$ . b. The image of $U$ and $J$ are the same distance along the same ray from $T$ . c. The image of $U$ and $J$ will not coincide after reflection over $S T$ . d. Line ST is the perpendicular bisector of the segment connecting $U$ and $J$ , because the perpendicular bisector is determined by 2 points that are both equidistant from the endpoints of a segment.

Studdy Solution

STEP 1

What is this asking? Why does flipping triangle $STU$ over line $ST$ land point $U$ right on top of point $J$ ? Watch out! Don't get tricked by fancy words like "coincide" – it just means "land on the same spot"!
Also, remember how reflections actually work; it's like mirroring an image.

STEP 2

1. Understand Reflections
2. Apply to the Problem
3. Analyze the Options

STEP 3

Imagine a mirror!
When you look in a mirror, your reflection appears on the other side, the same distance away.
That's what a reflection across a line does to a point.

STEP 4

The line of reflection acts like the mirror.
The original point and its reflection are equidistant from the line, meaning they're the same distance away.
Also, the line connecting the point and its reflection is perpendicular to the line of reflection, meaning it makes a right angle (90 degrees).

STEP 5

We're reflecting point $U$ across line $ST$ .
So, the image of $U$ , let's call it $U'$ , will be on the other side of $ST$ , the same distance away, and the line connecting $U$ and $U'$ will be perpendicular to $ST$ .

STEP 6

Now, look closely at the diagram (not provided here, but the student should have it).
Does it look like $ST$ is the perpendicular bisector of the segment $UJ$ ?
If it is, that means $J$ is the same distance from $ST$ as $U$ is, and the line $UJ$ is perpendicular to $ST$ .
This is exactly what we need for $J$ to be the reflection of $U$ !

STEP 7

This option talks about angles.
While angles are important in geometry, they aren't the key to understanding reflections in this case.
We need to focus on distances and perpendicularity.

STEP 8

This option talks about the image of $U$ and $J$ being the same distance along the same ray from $T$ .
This isn't what defines a reflection.
Reflections deal with distances from the line of reflection, not distances along a ray.

STEP 9

This option flat-out says the image of $U$ and $J$ won't coincide.
But we just figured out they should!
So, this is definitely wrong.

STEP 10

This option says line $ST$ is the perpendicular bisector of segment $UJ$ .
This is exactly what we were looking for!
If $ST$ is the perpendicular bisector of $UJ$ , then $J$ is the reflection of $U$ across $ST$ .
This is because the perpendicular bisector is made up of points that are equidistant from the endpoints of a segment.
Since $S$ and $T$ are both on the perpendicular bisector, they must be equidistant from $U$ and $J$ .

STEP 11

The correct answer is (d).
Line $ST$ is the perpendicular bisector of the segment connecting $U$ and $J$ , which means $J$ is the reflection of $U$ across $ST$ .

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Studdy Solution

STEP 1

What is this asking? Why does flipping triangle $STU$ over line $ST$ land point $U$ right on top of point $J$ ? Watch out! Don't get tricked by fancy words like "coincide" – it just means "land on the same spot"!
Also, remember how reflections actually work; it's like mirroring an image.

STEP 2

1. Understand Reflections
2. Apply to the Problem
3. Analyze the Options

STEP 3

Imagine a mirror!
When you look in a mirror, your reflection appears on the other side, the same distance away.
That's what a reflection across a line does to a point.

STEP 4

The line of reflection acts like the mirror.
The original point and its reflection are equidistant from the line, meaning they're the same distance away.
Also, the line connecting the point and its reflection is perpendicular to the line of reflection, meaning it makes a right angle (90 degrees).

STEP 5

We're reflecting point $U$ across line $ST$ .
So, the image of $U$ , let's call it $U'$ , will be on the other side of $ST$ , the same distance away, and the line connecting $U$ and $U'$ will be perpendicular to $ST$ .

STEP 6

STEP 7

This option talks about angles.
While angles are important in geometry, they aren't the key to understanding reflections in this case.
We need to focus on distances and perpendicularity.

STEP 8

This option talks about the image of $U$ and $J$ being the same distance along the same ray from $T$ .
This isn't what defines a reflection.
Reflections deal with distances from the line of reflection, not distances along a ray.

STEP 9

This option flat-out says the image of $U$ and $J$ won't coincide.
But we just figured out they should!
So, this is definitely wrong.

STEP 10

STEP 11

The correct answer is (d).
Line $ST$ is the perpendicular bisector of the segment connecting $U$ and $J$ , which means $J$ is the reflection of $U$ across $ST$ .

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Studdy Solution

STEP 1

What is this asking? Why does flipping triangle STUSTUSTU over line STSTST land point UUU right on top of point JJJ? Watch out! Don't get tricked by fancy words like "coincide" – it just means "land on the same spot"!Also, remember how reflections actually work; it's like mirroring an image.

STEP 2

1. Understand Reflections2. Apply to the Problem3. Analyze the Options

STEP 3

Imagine a mirror!When you look in a mirror, your reflection appears on the other side, the same distance away.That's what a reflection across a line does to a point.

STEP 4

STEP 5

We're reflecting point UUU across line STSTST.So, the image of UUU, let's call it U′U'U′, will be on the other side of STSTST, the same distance away, and the line connecting UUU and U′U'U′ will be perpendicular to STSTST.

STEP 6

STEP 7

This option talks about angles.While angles are important in geometry, they aren't the key to understanding reflections in this case.We need to focus on distances and perpendicularity.

STEP 8

This option talks about the image of UUU and JJJ being the same distance along the same ray from TTT.This isn't what defines a reflection.Reflections deal with distances from the *line of reflection*, not distances along a ray.

STEP 9

This option flat-out says the image of UUU and JJJ *won't* coincide.But we just figured out they should!So, this is definitely wrong.

STEP 10

STEP 11

The correct answer is (d).Line STSTST is the perpendicular bisector of the segment connecting UUU and JJJ, which means JJJ is the reflection of UUU across STSTST.

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

What is this asking? Why does flipping triangle $STU$ over line $ST$ land point $U$ right on top of point $J$ ? Watch out! Don't get tricked by fancy words like "coincide" – it just means "land on the same spot"!
Also, remember how reflections actually work; it's like mirroring an image.

1. Understand Reflections
2. Apply to the Problem
3. Analyze the Options

Imagine a mirror!
When you look in a mirror, your reflection appears on the other side, the same distance away.
That's what a reflection across a line does to a point.

We're reflecting point $U$ across line $ST$ .
So, the image of $U$ , let's call it $U'$ , will be on the other side of $ST$ , the same distance away, and the line connecting $U$ and $U'$ will be perpendicular to $ST$ .

This option talks about angles.
While angles are important in geometry, they aren't the key to understanding reflections in this case.
We need to focus on distances and perpendicularity.

This option talks about the image of $U$ and $J$ being the same distance along the same ray from $T$ .
This isn't what defines a reflection.
Reflections deal with distances from the line of reflection, not distances along a ray.

This option flat-out says the image of $U$ and $J$ won't coincide.
But we just figured out they should!
So, this is definitely wrong.

The correct answer is (d).
Line $ST$ is the perpendicular bisector of the segment connecting $U$ and $J$ , which means $J$ is the reflection of $U$ across $ST$ .