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

#1**+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....

CPhill
Feb 5, 2018

#2**+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}

CPhill
Feb 6, 2018

#4**+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

CPhill
Feb 6, 2018

#5**+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

CPhill
Feb 6, 2018

#8**+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

CPhill
Feb 6, 2018

#10**+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

CPhill
Feb 6, 2018