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1) a.                     In acute triangle ABC, we know AB = 7, BC = 8, and that Line{CA} is the shortest side. What is the smallest possible integer value of CA?

 

b.                  In obtuse triangle ABC, we know AB = 7, BC = 8, and that Line{CA} is the longest side. What is the smallest possible integer value of CA?

 

 

2)  a.          Two diagonals of a parallelogram have lengths 6 and 8. What is the largest possible length of the shortest side of the parallelogram?

 

b.             Two sides of an acute triangle are 8 and 15. How many possible lengths are there for the third side if it is an integer?

 

 

 

Thank You!smiley

 

 

P.s -- this Homework is due in some time

Guest Feb 9, 2018
edited by Guest  Feb 9, 2018
 #1
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1) a.  In acute triangle ABC, we know AB = 7, BC = 8, and that Line{CA} is the shortest side. What is the smallest possible integer value of CA?

 

The angle opposite  the longest side must be < 90°

 

Therefore

 

√[ CA^2  + 7^2 ] > 8^2

 

CA^2  + 49  > 64

 

CA^2  > 15

 

CA > 3.8

 

So....the smallest possible integer value of CA  is  4

 

 

b.  In obtuse triangle ABC, we know AB = 7, BC = 8, and that Line{CA} is the longest side. What is the smallest possible integer value of CA?

 

We must  have  that

 

AB^2  + BC^2  <  CA^2

 

7^2  +  8^2  <  CA^2

 

49  +  64  <  CA^2

 

113 < CA^2        take the square root of both sides

 

10.63  < CA   ⇒   CA  > 10.3

 

So....the shortest  integer side length for CA  is   11

 

 

2)  a.Two diagonals of a parallelogram have lengths 6 and 8. What is the largest possible length of the shortest side of the parallelogram?

 

The diagonals will bisect each other.....so we have a triangle with sides of  3 and 4

 

The largest possible (integer) length of the shortest side, S, of the parallellogram  must be

 

3 + S  >  4

 

S  >  1

 

So...the largest possible (integer) length of the shortest side  is  2

 

 

b.  Two sides of an acute triangle are 8 and 15. How many possible lengths are there for the third side if it is an integer?

 

Let S be the missing side...so we have....

 

S + 8 >  15

 

S >  7  so the shortest integer length is 8

 

But also...since it's an acute triangle,  the longest possible side is

 

√ [ 8^2 + 15^2 ]  > S

 

17 >  S  ⇒   S <  17

 

So....the greatest possible integer  length of S  is  16

 

So...the number of possible lengths  is   16  - 8 +  1      =  9

 

 

 

cool cool cool

CPhill  Feb 9, 2018

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