Geometry Forum - Problem of the Week
Solutions - Minimum Perimeter of an Inscribed Triangle, Jan. 9-13, 1995 Annie says:
Very nice job by these folks this week. As I said in the problem, this problem was stuck on a door here in the math department, and we had played around with it a bit. Special kudos to Sean this week, as he gives a very nice explanation of both parts, without prodding from me, and even includes the idea of a "limit", or "lower bound", in his answer. Good work!
Sean Nichols
Hello. This is Sean Nichols at Burnaby South Secondary School, in Burnaby, BC, Canada. My solution for the Problem of the Week for January 9-13 is as follows:For the smallest triangle: As any two sides of the triangle get closer and closer, the triangle itself becomes closer and closer to a line. Any two sides of a triangle are longer than the third, so the perimeter of a triangle cannot be shorter than twice the langth of any one of its sides. Therfore, the length of a line formed by closing in any two sides (or even 2x the line) is smaller than the total perimeter of any triangle. So to get the smallest possible triangle, we need to close in any two sides so that they are as close together as possible. Since the triangle _must_ contain the center of the circle, these two lines will have to be on either side of this center. that means that the third side, being as close to 0 as possible, can be considered as having effectively no length, and the two remaining sides lie along the diameter. This means that the perimeter of the triangle is for all purposes twice the diameter, or four times the radius. This is equal to 4 (the circle is defined as a unit circle). However, since this would have the length of the third side as 0, this is impossible, and the perimeter's _lower boundary_ is 4. So the perimeter will have the smallest length possible larger than 4.
For the largest triangle: Because the perimeter of the triangle becomes less as any two sides become closer together, therefore it must necessarily get larger as all sides get farther apart, i.e: the angles between all sides are as large as possible. This state is achieved when the triangle is equilateral, as all angles in an equilateral triangle are 60 degrees, and if the size of any angle is increased any more, then the size of at least one other angle is decreased, and the sides that include it become closer together. As already stated, this makes the perimeter smaller. So, if we have an equilateral triangle, we will have the following figure:
.
. /|\ .
. a|b .
. / | \ .
. | .
. / o \ .
. | .
. / | \ .
._______________.
. .
.
The triangle is equilateral, so m<a + m<b = 60. An equilateral triangle is isosceles, with the diameter as a line of symmetry, so m<a = m<b = 1/2 60 = 30. The length of the side on the left can be found by trigonometry to be 2cos(30). The total perimeter is three times this, of 6cos(30) (roughly equal to 5.196).
Thank You,
- Sean Nichols
Kim Carbone
Kim Carbone, grade 9, Fairfield HS
Annie,
The triangle with a perimeter of 4 is the smallest triangle because it has the least
area, just over zero.By drawing triangles, you can see the pattern of the base and
height so that the smallest triangle is one with a base just over zero and a height
just under 2. The triangle with the largest area is an equilateral triangle.
If the circle's radius is equal to 1, then the diameter is equal to 2. Therefore, if a
triangle is to be drawn containing the center, the sides are x, y, and z.
The perimeter for the smallest triangle would be a little above 4. The reason being,
if you draw a line through the center of the circle (the diameter), it equals 2. If you
draw another line very close to the first one, it would measure a little less than
two. Together they would equal a little less than four. The third side would have
to be more than the difference of the first two sides otherwise it would not make a
triangle. There, technically, the smallest triangle would be a little more than 4.
The perimeter of the largest triangle is 5.25 inches and the perimeter of the
smallest triangle is the smallest number after 4 inches. I found this by drawing a
circle of one inch radius. Then I took my ruler and started drawing different
triangles and determining which one was the largest and smallest by using the
area of a triangle, 1/2 bh. Then after drawing many triangles, I found which ones
were the greatest and smallest and measured their bases and legs and found the
perimeter.
The triangle with a perimeter of 4 is the smallest triangle because it has the least
area, just over zero.By drawing triangles, you can see the pattern of the base and
height so that the smallest triangle is one with a base just over zero and a height
just under 2. The triangle with the largest area is an equilateral triangle.
The biggest would be an equilateral triangle because it's verticies have to be on
the circle, and it has to contain the center. The trianlge is trying to "push" it's way
out of the circle , so it's sides are expanded equally to become the biggest triangle
it can be.
I figure that the smallest will have to look more like a wedge of pie, than an
obtuse triangle with the diameter of the circle being the hypotenuse. Because if
you take the radii, which are 2 of the sides of the triangle you know they will
always stay the same length. So, you have to be smart with the third leg. (Make it
as small as possible!)
Now that I know what it will look like, to make the smallest triangle you will have
to use the smallest central angle possible. which might be too small to draw, so
you don't know the perimeter. (they never said it had to be drawable) (maybe it's
just too small for the naked eye)
I think that the triangle with the largest perimeter is an equilateral triangle
touching the parts of a ciccle including the center. The reasoning is that all three
sides of the triangle are quite long, not just two long and one short. another
argument is that the perimeter is 5.16 a lot bigger than 4 describes later. I used sin,
cos, and tan to figure this out. Also, it goes closest to representing the whole
circle's circumference . Ther are 3 equal chords.
The triangle with the smallest perimeter is one that has one side the diameter, the
other at the smallest length possible and the third side just connects the two sides.
This is so because then the third side is very, very short even though the other
two sides are long. I think the perimeter of this triangle must be about 4 because
the line going through the center must be 2, the one real close to it a little, very
small amount, less than 2, then the third side is very small making the 2nd and 3rd
side about 2. That adds up to 4.
My answer is that the smallest perimeter is only just above 4. In this triangle one
side goes through the center, the second side goes almost straight up, making the
last side something like .01 long. The biggest triangle's perimeter is an equilateral
triangle in the circle. It's perimeter is about 5.25. The way I got this is that I
measured each side and it was about 1.75.
[Susie got the smallest triangle correct, but not the largest. But I liked her
explanation of the smallest, so I've included her whole response here.]
I saw the problem as being as follows - what are the smallest and the biggest
triangles you can fit inside a circle where all three vertices touch the
outside of the circle, and the center of the circle is within the triangle.
I began by drawing a circle with a diameter of 27/8 inches. Inside that circle I
drew an equilateral triangle , ABC. The center of the circle is also the center of the
triangle. Each side of triangle ABC equals 2 14/16 inches. The perimeter of ABC
equals 8 5/8 inches. The next triangle I drew I called DBE. This trianlge was very
long and narrow . It was an isosceles with vertex angle DBE. This triangle just
barely contains the center of the circle. It is so thin that the perimeter can be
considered very basiclly 2D. So the perimeter of triangle DBE is 6 6/8 inches. The
final triangle I drew within my circle I labled FBG. the base of this triangle, FG,
runs right along the center point of the circle. so it cuts my circlein half. FG equals
the diameter of my circle and the other two sides equals 2 1/2 inches. So th e
perimeter of this triangle equals 8 3/8 inches.
What I am able to conclude from the process i just wrote about is the following.
Triangle DBE is roughly the closest I could either side of the triangle to the center
of the circle without a side hitting the center. Triangle FBG is the triangle with the
largest base possible, being because its base is the complete diameter of the circle.
I know that the largest and the smallest triangles have to have two of their
vertices between arcs, FD and GE. Because the bases of isosceles triangles DBE
and FBG and the largest and smallst bases possible.
side x<=2
side y<2
side z: 0
Julie Black
Julie Black, grade 9, Fairfield HS
Ashleigh Rossman
Ashleigh Rossman
10th grade
College Park High School, Pleasant Hill, California
Faaiza Bashir
Faaiza Bashir
9th grade
College Park High School, Pleasant Hill, California
Roy Holtman
Roy Holtman
9th grade
College Park High School, Pleasant Hill, California
Susie Goetz
Susie Goetz
9th grade
College Park High School, Pleasant Hill, California
Previous page ||
Next problem ||
Previous problem ||
Table of Contents ||
Forum Home Page