Q14. Given: PA炉鈯ヂ爌lane X; QB炉鈯�... [FREE SOLUTION]

91影视

Geometry

Ray C. Jurgensen, Richard G. Brown, John W. Jurgensen

$Math Studyset 91影视 Explanations$ Math

Student Edition 路 227 pages

ISBN: 9780395977279

Chapter 4: Q14. (page 132)

Given: $\overset{炉}{P A} 鈯�$ plane $X$ ; $\overset{炉}{Q B} 鈯�$ plane $X$ ; $O$ is the midpoint of $\overset{炉}{A B}$
Prove: $O$ is the midpoint of $\overset{炉}{P Q}$

Short Answer

Expert verified

Statement	Reason
1. $m 鈭� P A O = m 鈭� Q B O = 90 掳$	From given
2. $\overset{炉}{A O} 鈮� \overset{炉}{B O}$	From given
3. $鈭� P O A 鈮� 鈭� Q O B$	Vertically opposite angles
4. $螖 P A O 鈮� 螖 Q B O$	ASA Congruency rule
5. $\overset{炉}{P O} 鈮� \overset{炉}{Q O}$	corresponding parts of congruent triangles are congruent
6. $O$ is midpoint of $\overset{炉}{P Q}$	Definition of midpoint

Step by step solution

Step 1. Show that ΔPAO≅ΔQBO.

Using given $m 鈭� P A O = m 鈭� Q B O = 90 掳$

Since, midpoint divides line segment into two congruent line segments

So, $\overset{炉}{A O} 鈮� \overset{炉}{B O}$

Also, $鈭� P O A and 鈭� Q O B$ forms a pair of vertically opposite angles, so they are congruent

Thus, $螖 P A O 鈮� 螖 Q B O$ by ASA (Angle-Side-Angle) congruency criteria

Step 2. Show that PO¯≅QO¯.

Since, corresponding parts of congruent triangles are congruent

So, $\overset{炉}{P O} 鈮� \overset{炉}{Q O}$

Step 3. Show that is midpoint of PQ¯

By definition of midpoint, midpoint divides line segment into two congruent line segments

Since, $\overset{炉}{P O} 鈮� \overset{炉}{Q O}$

Thus, width="14" height="19" role="math">O is midpoint of $\overset{炉}{P Q}$

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91影视

Given: $\overset{炉}{P A} 鈯�$ plane $X$ ; $\overset{炉}{Q B} 鈯�$ plane $X$ ; $O$ is the midpoint of $\overset{炉}{A B}$
Prove: $O$ is the midpoint of $\overset{炉}{P Q}$

Short Answer

Step by step solution

Step 1. Show that ΔPAO≅ΔQBO.

Step 2. Show that PO¯≅QO¯.

Step 3. Show that is midpoint of PQ¯

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91影视

Given: PA炉鈯�plane X; QB炉鈯�plane X; Ois the midpoint of AB炉Prove:Ois the midpoint ofPQ炉

Short Answer

Step by step solution

Step 1. Show that ΔPAO≅ΔQBO.

Step 2. Show that PO¯≅QO¯.

Step 3. Show that is midpoint of PQ¯

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Theoretical and Mathematical Physics

Mechanics Maths

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Calculus

Logic and Functions

Study anywhere. Anytime. Across all devices.

Given: $\overset{炉}{P A} 鈯�$ plane $X$ ; $\overset{炉}{Q B} 鈯�$ plane $X$ ; $O$ is the midpoint of $\overset{炉}{A B}$
Prove: $O$ is the midpoint of $\overset{炉}{P Q}$