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1)The perimeter of an isosceles triangle is 36 centimeters and two sides of the triangle are in the ratio 2:5. What is the number of centimeters in the length of the longest side?

 

2)In how many ways can we choose one number from the set {1,2,3}, one number from the set {4,5,6}, and one number from the set {7,8,9} such that the three could be the sides of a nondegenerate triangle?

 

3)A triangle with integer sides has perimeter 12. How many such non-congruent triangles are there? (A 3-4-5 triangle is considered congruent to a 3-5-4 triangle because we can reflect and rotate the triangles until they match up.)

 

4)The distance from Capital City to Little Village is 660 miles. From Capital City to Mytown is 310 miles, from Mytown to Yourtown is 200 miles, and from Yourtown to Little Village is 150 miles. How far is it from Mytown to Little Village?

 

5)A triangle has integer length sides. If two sides of the triangle are 16 and 21, how many possible lengths are there for the third side?

 

6)An obtuse triangle has integer length sides. If two sides of the triangle are 16 and 21, how many possible lengths are there for the third side?

 

7)Two altitudes of a triangle have lengths 12 and 14. What is the longest possible integer length of the third altitude?

FiestyGeco  Feb 5, 2018
edited by FiestyGeco  Feb 5, 2018
 #1
avatar+87691 
+1

1)The perimeter of an isosceles triangle is 36 centimeters and two sides of the triangle are in the ratio 2:5. What is the number of centimeters in the length of the longest side?

 

The sides must  be  in the ratio   2 : 5 : 5

 

So......each equal part must be   3 cm

 

So....the number of cm in the longest side =

 

5 / 12  *  36  = 

 

5 * (36/12) =

 

5 * 3  =

 

15 cm

 

The shortest side  is  (2/12) (36)  =  6

 

So   15 + 15 + 6  = 36

 

EDit to correct a previous error....

 

 

cool cool cool

CPhill  Feb 5, 2018
edited by CPhill  Feb 6, 2018
 #3
avatar
0

1). CPhill: Did you make a mistake in calculating the sides of the isosceles triangle? 9 + 9 + 20 =38??

Is it 5: 5: 2, or 5/12 x 36 =15, 15, 6 =36 cm??

Guest Feb 6, 2018
 #6
avatar+87691 
0

 

 

 

cool cool cool

CPhill  Feb 6, 2018
edited by CPhill  Feb 6, 2018
 #7
avatar
0

But is it allowed? 8+8 < 20 ??!!

Guest Feb 6, 2018
 #9
avatar+87691 
0

See my corrected answer.....

 

cool cool cool

CPhill  Feb 6, 2018
 #2
avatar+87691 
+1

2)In how many ways can we choose one number from the set {1,2,3}, one number from the set {4,5,6}, and one number from the set {7,8,9} such that the three could be the sides of a nondegenerate triangle?

 

We can  eliminate 1 as one of the sides because  1 + 6  = 7 

But this sum must exceed 7

 

 

So...the possibilities are

 

{2, 6, 7 }

{3, 5, 7 }

{3, 6, 7}

{3, 6, 8}

 

 

cool cool cool

CPhill  Feb 6, 2018
 #4
avatar+87691 
+1

3)A triangle with integer sides has perimeter 12. How many such non-congruent triangles are there? (A 3-4-5 triangle is considered congruent to a 3-5-4 triangle because we can reflect and rotate the triangles until they match up.)

 

 

Possibilities :

 

3 - 4 - 5

4 - 4 - 4

5 - 5 - 2

 

 

cool cool cool

CPhill  Feb 6, 2018
 #5
avatar+87691 
+1

4)The distance from Capital City to Little Village is 660 miles. From Capital City to Mytown is 310 miles, from Mytown to Yourtown is 200 miles, and from Yourtown to Little Village is 150 miles. How far is it from Mytown to Little Village?

 

Here's the layout [  assuning that all the cities are collinear ]

 

 

C                  660                                              L

 

C       310             M

 

                            M     200       Y 

 

                                                  Y      150       L

 

 

Mytown to Little Village  =  200 + 150    =  350 miles

 

 

 

cool cool cool

CPhill  Feb 6, 2018
 #8
avatar+87691 
+1

5)A triangle has integer length sides. If two sides of the triangle are 16 and 21, how many possible lengths are there for the third side?

 

We have to have that

 

If the missing side is the shortest side  then

Shortest side + 16 > 21

So...the shortest side > 5

 

And the longest side must be

16 + 21 > longest side

Longest side < 37

 

So....the possible sides are

 

5 < possible sides < 37               }

 

So the possible sides are   36 - 6 + 1     =   31 possible lengths

 

 

cool cool cool

CPhill  Feb 6, 2018
edited by CPhill  Feb 6, 2018
 #10
avatar+87691 
+1

6)An obtuse triangle has integer length sides. If two sides of the triangle are 16 and 21, how many possible lengths are there for the third side?

 

Let  M be the missing side

 

For  M to be a possible solution, we must have that

 

M^2  > 16^2 + 21^2  

 

M^2 > 697      

 

M > ≈ 26.4

 

So  the shortest integer length for M  is 27

 

But...

 

21 + 16 > M   which means that  M < 37

 

So...the longest integer length for M   is 36

 

So...there are  36 - 27 + 1   = 10 possible side lengths

 

 

cool cool cool

CPhill  Feb 6, 2018

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