Two lines m and n intersect at point P, forming a 40^∘ angle. a. You reflect point B across line

Name: Problem 4, Chapter 7: Discovering Geometry an Investigative Approach
Uploaded: 2021-01-15T16:43:31-08:00
Duration: 1 min 1 s
Channel: Nick Johnson

step 1 Okay, so here is our diagram. Now, because we know that angle P here is 40 degrees, each reflect

Question

Two lines $m$ and $n$ intersect at point $P,$ forming a $40^{\circ}$ angle. a. You reflect point $B$ across line $m$, then reflect the image of $B$ across line $n .$ What angle of rotation about point $P$ rotates the second image of point $B$ back to its original position? b. What if you reflect $B$ first across $n$, and then reflect the image of $B$ across $m ?$ Find the angle of rotation that rotates the second image back to the original position. (FIGURE CAN NOT COPY)

   Two lines $m$ and $n$ intersect at point $P,$ forming a $40^{\circ}$ angle.
a. You reflect point $B$ across line $m$, then reflect the image of $B$ across line $n .$ What angle of rotation about point $P$ rotates the second image of point $B$ back to its original position?
b. What if you reflect $B$ first across $n$, and then reflect the image of $B$ across $m ?$ Find the angle of rotation that rotates the second image back to the original position.
(FIGURE CAN NOT COPY)

Discovering Geometry an Investigative Approach

Michael Serra 4th Edition

Chapter 7, Problem 4

Instant Answer

Solved by Expert Nick Johnson

01/15/2021

Step 1

When we reflect point $B$ across line $m$, it rotates by $40^{\circ}$ about point $P$. Show more…

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Two lines $m$ and $n$ intersect at point $P,$ forming a $40^{\circ}$ angle. a. You reflect point $B$ across line $m$, then reflect the image of $B$ across line $n .$ What angle of rotation about point $P$ rotates the second image of point $B$ back to its original position? b. What if you reflect $B$ first across $n$, and then reflect the image of $B$ across $m ?$ Find the angle of rotation that rotates the second image back to the original position. (FIGURE CAN NOT COPY)

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Key Concepts

Composition of Reflections

The composition of two reflections is a geometric transformation obtained by performing one reflection followed by another. When the reflection axes intersect, this composition is equivalent to a rotation about the point of intersection, with the rotation angle being twice the measure of the angle between the reflecting lines.

Reflection

Reflection is a geometric transformation that produces a mirror image of a figure over a line known as the axis of reflection. This transformation preserves distances and angles, meaning that the original shape and its reflected image are congruent even though their orientations differ.

Rotation

Rotation is a transformation that turns a figure about a fixed point, known as the center of rotation, by a certain angle. In a rotation, every point of the original figure moves along a circular arc centered on the rotation point, preserving the shape’s size and form.

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