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Elementary Geometry For College St...

7th Edition
Alexander + 2 others
ISBN: 9781337614085

Solutions

Chapter
Section
BuyFindarrow_forward

Elementary Geometry For College St...

7th Edition
Alexander + 2 others
ISBN: 9781337614085
Textbook Problem

In Exercises 13 to 16, provide the missing reasons.

Given: Δ A B C ; M and  N are midpoints of A B ¯ and A C ¯ , respectively

Prove: Δ A M N Δ A B C

PROOF
Statements Reasons
1. Δ A B C ; M and  N are the midpoints of A B ¯ and A C ¯ , respectively. 1. ?
2. A M = 1 2 ( A B ) and
A N = 1 2 ( A C )
2. ?
3. M N = 1 2 ( B C ) 3. ?
4. A M A B = 1 2 , A N A C = 1 2 ,
and  M N B C = 1 2
4. ?
5. A M A B = A N A C = M N B C 5.?
6. Δ A M N Δ A B C 6. ?

To determine

To provide:

The missing reasons.

Explanation

Given:

Given: ΔABC; M and N are midpoints of AB¯ and AC¯, respectively.

Prove: ΔAMNΔABC

PROOF
Statements Reasons
1. ΔABC;Mand N are midpoints of AB¯ and AC¯, respectively. 1. ?
2. AM=12(AB) and
AN=12(AC)
2. ?
3. MN=12(BC) 3. ?
4. AMAB=12,ANAC=12,
and MNBC=12
4. ?
5. AMAB=ANAC=MNBC 5.?
6. ΔAMNΔABC 6. ?

Definition:

SSS:

If the three sides of one triangle are proportional (in length) to the three corresponding sides of a second triangle, then the triangle are similar.

Description:

Given that ΔABC;Mand N are midpoints of AB¯ and AC¯, respectively.

To Prove: ΔAMNΔABC

From the given figure, it is observed that the line segment MN joining the two sides of AB and AC. In the triangle ABC, M is the midpoint of AB and N is the midpoint of AC.

By the definition of midpoint, AM=12AB and AN=12AC.

It is known that the line segment MN joins the midpoints of two sides a triangle, its length MN is half the length of BC

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