Book edition Student Edition

Author(s) Ray C. Jurgensen, Richard G. Brown, John W. Jurgensen

Pages 227 pages

ISBN 9780395977279

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Short Answer

A rolling ball that does not have much spin will bounce off a wall so that the two angles that the path forms with the wall are congruent. Thus, to roll the ball from B off the wall shown and into hole H, you need to aim the ball so that $∠ 1 ≅ ∠ 2$ .
Let $H'$ be the image of H by reflection in the wall. $\vec{B H'}$ Intersects the wall at P. Why is $∠ 1 ≅ ∠ 3$ ? Why is $∠ 3 ≅ ∠ 2$ ? Why is $∠ 1 ≅ ∠ 2$ ? You can conclude that if you aim for $H'$ , the ball will roll to H.
Show that the distance travelled by the ball equals the distance H.

$∠ 1 ≅ ∠ 2, ∠ 1 ≅ ∠ 3 and ∠ 3 ≅ ∠ 2$ .
The distance travelled by the ball equals the distance $B H'$ .

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

a.Step 1. Given Information.

$H'$ Is the image of H by reflection in the wall and $\bar{B H'}$ intersects the wall at P.

Step 2. Proof.

$∠ 1 ≅ ∠ 3 [vertically opposite angles]$

In triangles $P X H and P X H'$ .

$\begin{matrix} P X = P X [common side] \\ m ∠ X = m ∠ X [each equals to 90 °] \\ H X = H X' [perpendicular bisector] \end{matrix}$

By side angle side congruency

$\begin{matrix} Δ P X H ≅ Δ P X H' \\ ∠ 2 ≅ ∠ 3 \end{matrix}$

Since $∠ 1 ≅ ∠ 3 and ∠ 2 ≅ ∠ 3$ therefore $∠ 1 ≅ ∠ 2$

Step 3. Conclusion.

Hence, the congruency of angles is proved.

b.Step 1. Given Information.

$H'$ Is the image of H by reflection in the wall and $\bar{B H'}$ intersects the wall at P.

Step 2. Proof.

$\begin{matrix} Δ P X H ≅ Δ P X H' \\ P H = P H' \end{matrix}$

Distance travelled by the ball is,

$\begin{matrix} B P + P H = B P + P H' \\ = B H' \end{matrix}$

Step 3. Conclusion.

Hence, it is proved that the distance travelled by the ball equals the distance $B H'$ .

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Short Answer

Step by Step Solution

a.Step 1. Given Information.

Step 2. Proof.

Step 3. Conclusion.

b.Step 1. Given Information.

Step 2. Proof.

Step 3. Conclusion.

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Step by Step Solution

a.Step 1. Given Information.

Step 2. Proof.

Step 3. Conclusion.

b.Step 1. Given Information.

Step 2. Proof.

Step 3. Conclusion.

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