Math Is Fun Forum

math137

Member

Registered: 2014-08-07

Posts: 12

Geo Problems

Here's a couple problems that I'm stuck on. It would be appreciated if someone could help.

1. One side of a triangle is 10. What is the smallest possible perimeter of the triangle, if it is known to be a positive integer?

2. A (non-degenerate) triangle with integer sides has perimeter 12. How many such non-congruent triangles are there?

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

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. In the triangle shown, n is a positive integer, and angle A>angle B>angle C. How many possible values of n are there?

Thx!

Last edited by math137 (2014-08-07 14:14:56)

Bob

Administrator

Registered: 2010-06-20

Posts: 10,157

Re: Geo Problems

hi math137,

1. One side of a triangle is 10. What is the smallest possible perimeter of the triangle, if it is known to be a positive integer?

The other two sides must add up to more than 10, otherwise they won't meet to make the third vertex.

2. A (non-degenerate) triangle with integer sides has perimeter 12. How many such non-congruent triangles are there?

I cannot think of a better way than 'slogging through' all the ways of adding three numbers to 12, rejecting any that disobey the rule a + b > c .

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

Same approach here.

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?

I've met this one before.  The places appear to be in a straight line.

5. In the triangle shown, n is a positive integer, and angle A>angle B>angle C. How many possible values of n are there?

I think I've seen this one before too.

Once again you can use the property of any triangle that a + b > c where a, b, and c are the sides in any order. So you can make three inequalities.  Here's one:

3n + 1 + 4n - 9 > 3n + 4

You can also use the angle property to make inequalities like this

3n + 4 > 4n - 9

In each case, simplify and get a restriction on n.

By the time you have all of these, you'll have your answer.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

ElainaVW

Member

Registered: 2013-04-29

Posts: 580

Re: Geo Problems

Hello;

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Geo Problems

Hi EVW;

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

math137

Member

Registered: 2014-08-07

Posts: 12

Re: Geo Problems

I have made the three inequalities involving the three side lengths. From that, I get n>-7, n>3/2, n>3, so from this I conclude n>3. From the angle inequality you gave me, I got n<13 so shouldn't there be 17 different possible values? It says it's wrong though.

ElainaVW

Member

Registered: 2013-04-29

Posts: 580

Re: Geo Problems

Bob

Administrator

Registered: 2010-06-20

Posts: 10,157

Re: Geo Problems

hi math137

Those inequalities are correct but there is one missing.  17 wouldn't be right from 3 < n < 13.  There are 9 values in that range.  BUT .....

Because angle B > angle C => 3n+1 > 4n - 9

This leads to a lower upper bound.

You should then get the same number of n values as ElainaVW

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

math137

Member

Registered: 2014-08-07

Posts: 12

Re: Geo Problems

Ok thx guys!

Bob

Administrator

Registered: 2010-06-20

Posts: 10,157

Re: Geo Problems

You're welcome.    I made a small edit while you were typing.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

SolarDevil

Member

Registered: 2014-09-14

Posts: 37

Re: Geo Problems

Hello

I was looking at these set of posts but still don't get how to get number 3. Could you help?

Thanks
SolarDevil

---

Herro! Sycamore School will win National Science Bowl this year!!!

Bob

Administrator

Registered: 2010-06-20

Posts: 10,157

Re: Geo Problems

hi SolarDevil

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

If you had three rods to try and make a triangle, there are restrictions on the lengths that are possible.  Let's say the lengths are a, b and c with c being the longest.

If c > a + b then you won't be able to join the rods together because, even when a and b are placed in a straight line, a + b is not long enough to reach.

So c < 8 + 15.

Now suppose c is a shorter side.

Imagine that the 8 and 15 are joined and the 8 is folded back so that it lies on top of the 15.  The distance between the free ends is now 7.  Once again c will not be able to join these ends is it is less than 7, so we have

7 < c < 23

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

reaganchoi

Member

Registered: 2015-12-18

Posts: 2

Re: Geo Problems

I think you forgot the "acute" part.
If c was the longest side, then c<17 (because 8-15-17 is a Pythagorean Triple).
If c was one of the shorter sides, then it has to be 8^2+c^2>15^2
so 64+c^2>225
so c^2>161
so c>sqrt(161)
sqrt(161)<c<17
13, 14, 15, and 16 work.
(I might be wrong)

Last edited by reaganchoi (2015-12-18 00:25:43)

Relentless

Member

Registered: 2015-12-15

Posts: 631

Re: Geo Problems

Problem four did not seem too difficult. All you have to do is visualise a circular continuum of circles (Yourtown) within a circle (Little Village) around Capital City, and it's clear that if there's only one solution they're in a straight line. I just drew the problem and it was transparent that way. It is neat though.
Guess I will ponder three when I get some sleep xP

Edit: I guess if you saw that 310 + 200 + 150 = 660 that would make it transparent as well, haha. But that is why I need sleep!

Last edited by Relentless (2015-12-18 05:26:35)

Relentless

Member

Registered: 2015-12-15

Posts: 631

Re: Geo Problems

Basically you're right about problem three: Bob forgot the acute part of the definition. I just drew the triangles, since there aren't that many possibilities. Triangles where n is 8 to 12 or 18 to 22 are obtuse. 17 is a right triangle. 13 to 16 work so there are four triangles.