Q26. For Exercises 23-27 write proofs... [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: Q26. (page 158)

For Exercises 23-27 write proofs in paragraph form. (Hint: you can use theorems from this section to write fairly short proofs for exercise 23 and 24.)
Given: $m 鈭� RTS = 90$ ; $\overset{鈫�}{MN}$ is the $鈯�$ bisector of $\overset{炉}{TS}$ .
Prove: $\overset{炉}{TM}$ is a median.

Short Answer

Expert verified

It is being given that $m 鈭� RTS = 90$ ; $\overset{鈫�}{MN}$ is the $鈯�$ bisector of $\overset{炉}{TS}$ .

As, $\overset{鈫�}{MN}$ is the $鈯�$ bisector of $\overset{炉}{TS}$ , therefore $鈭� MNS = 90$ and $TN = NS$ .

In the triangles $螖 RTS$ and $螖 MNS$ , it can be noticed that:

$鈭� RTS = 鈭� MNS = 90$

$鈭� RST = 鈭� MSN (common)$

Therefore, the triangles $螖 RTS$ and $螖 MNS$ are similar triangles.

In the similar triangles, the corresponding sides are in proportion.

Therefore, it can be obtained that:

$\frac{RS}{MS} = \frac{TS}{NS} \frac{RS}{MS} = \frac{TN + NS}{NS} \frac{RS}{MS} = \frac{NS + NS}{NS} \frac{RS}{MS} = \frac{2 NS}{NS} \frac{RS}{MS} = 2 RS = 2 MS RM + MS = 2 MS RM = 2 MS - MS RM = MS$

As $RM = MS$ , therefore it can be obtained that $M$ is the midpoint of $TS$ .

Therefore, $TM$ is the median.

Hence proved.

Step by step solution

Step 1. Observe the given diagram.

The given diagram is:

Step 2. Description of step.

It is being given that $m 鈭� R T S = 90$ ; $\overset{鈫�}{M N}$ is the $鈯�$ bisector of $\overset{炉}{T S}$ .

As, $\overset{鈫�}{M N}$ is the $鈯�$ bisector of $\overset{炉}{T S}$ , therefore $鈭� M N S = 90$ and $T N = N S$ .

In the triangles $螖 R T S$ and $螖 M N S$ , it can be noticed that:

$鈭� R T S = 鈭� M N S = 90$

$鈭� R S T = 鈭� M S N (common)$

Therefore, the triangles $螖 R T S$ and $螖 M N S$ are similar triangles.

Step 3. Description of step.

In the similar triangles, the corresponding sides are in proportion.

Therefore, it can be obtained that:

$\frac{R S}{M S} = \frac{T S}{N S} \frac{R S}{M S} = \frac{T N + N S}{N S} \frac{R S}{M S} = \frac{N S + N S}{N S} \frac{R S}{M S} = \frac{2 N S}{N S} \frac{R S}{M S} = 2 R S = 2 M S R M + M S = 2 M S R M = 2 M S 鈭� M S R M = M S$

As $R M = M S$ , therefore it can be obtained that $M$ is the midpoint of $R S$ .

Therefore, $T M$ is the median.

Hence proved.

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91影视

For Exercises 23-27 write proofs in paragraph form. (Hint: you can use theorems from this section to write fairly short proofs for exercise 23 and 24.)
Given: $m 鈭� RTS = 90$ ; $\overset{鈫�}{MN}$ is the $鈯�$ bisector of $\overset{炉}{TS}$ .
Prove: $\overset{炉}{TM}$ is a median.

Short Answer

Step by step solution

Step 1. Observe the given diagram.

Step 2. Description of step.

Step 3. Description of step.

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91影视

For Exercises 23-27 write proofs in paragraph form. (Hint: you can use theorems from this section to write fairly short proofs for exercise 23 and 24.)Given: m鈭�RTS=90;MN鈫� is the鈯� bisector of TS炉.Prove:TM炉is a median.

Short Answer

Step by step solution

Step 1. Observe the given diagram.

Step 2. Description of step.

Step 3. Description of step.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Statistics

Mechanics Maths

Theoretical and Mathematical Physics

Decision Maths

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

For Exercises 23-27 write proofs in paragraph form. (Hint: you can use theorems from this section to write fairly short proofs for exercise 23 and 24.)
Given: $m 鈭� RTS = 90$ ; $\overset{鈫�}{MN}$ is the $鈯�$ bisector of $\overset{炉}{TS}$ .
Prove: $\overset{炉}{TM}$ is a median.