Correct solutions were submitted by:
College Park High School, Pleasant Hill, California
Philip Horner, Asa Sharma, Angie Bush, Shastina Larson,
Rehecca Pearson, Grade 9
Fairfield High School, Fairfield, Connecticut
Carly Tubbs, Grade 9
Granada High School, Livermore, California
Margaret Sharp, Ethan Castor, Mike Sue, Neil Tucker,
Deanne Derego, Grade 10
Carolyn Watling, Grade 9
Hanover High School, Hanover, New Hampshire
Carson Henry, Grade 9
Hine Junior High, Washington, DC
Someone, Grade ?
Livermore High, Livermore, California
Brooke Freeman, Grade 10
Maret School, Washington, DC - 8th and 9th grade students
Lauren Austrian, Rachel Block, Keira Cohen, Marie
Ghadar, Ann Jackson, Philip Kowalczyk, Julia Lipton,
Jenny Lunstead, Peter Miller, Justin Schauer, Peter
Shattuck, Lee Teslik, James Whittle, Matt Winn
Mt. St. Joseph Academy, Flourtown, Pennsylvania
Jill Sommer, Michele Weiss, Katie Walder and Lindsay
Parsons, Sarah Joyce and Lauren Grabowski, Grade 10
Susan Tull and Jackie Mattera, Kelly Larkin and
Annie McIntyre, Liz Croney, Grade 9
Murray Junior High School, Ridgecrest, California
Thomas S. Kuo, Grade 7
Tasmania, Australia
Amy Forster, Age 11, Home Schooled
York High School, York, Canada
Kevin Scorza & Randy Kuczaj, Grade 10
Gary Boraas, Instructor, Lac qui Parle Valley High School,
Madison, Minnesota
From: Carson Henry
Grade: 9
School: Hanover High School
1. The first way to construct a isosceles triangle is to draw two 90
degree triangles with sides that are equal and then put them
together so that their bases are touching, and you have a isosceles
triangle.
2. The second way to construct an isosceles triangle is to construct
a rectangle. Then draw a line connecting the upper right hand
corner to the lower left hand corner- you will create two equal 90
degree triangles. Then flip them so the bases are touching and you
have an isosceles triangle.
3. The third way to construct a isosceles triangle is to construct a
trapezoid with equal sides (the sides that are slanted). Then draw a
line on both sides from the vertex of the sides and the upper base
straight down. It should create two equal 90 degree triangles. Then
take the two triangles that you created and flip them so their bases
are touching and you have an isosceles triangle.
************************************************
From: Brooke Freeman
Grade: 10
School: Livermore High
1) Use a compass to draw a circle with any radius. Draw any
diameter and use arcs to find the perpendicular bisector of that
segment. Continue the line until it reaches the circle. With those
three points draw the other two lines to that point.
2) Use a compass and draw a circle, draw any chord that you
please, bisect that chord, and connect the remaining two chords.
3) Use a compass and draw a circle, draw two perpendicular
diameters in the circle, use those four points to draw a square,
then draw either of the diagonals of that square.
************************************************
From: Margaret Sharp
Grade: 10
School: Granada
1. Draw a square and then draw in one of the square's diagonals.
You get two isosceles triangles because the sides of a square are the
same size.
2. Draw a rectangle and then draw in both of the diagonals. The
diagonals of a rectangle are congruent so you can form four
isosceles triangles.
3. Draw a circle, then find the center point. From this point draw
two diameters perpendicular to each other. Connect two adjacent
points together and continue around the circle. Four isosceles
triangles are made in this way because the two diameters form four
radii, all the same length. These form the legs. The four triangles
are also congruent because they all have two congruent legs and a
congruent angle in between these legs. The triangles can then form
a rhombus because all the hypotenuses are congruent, because
corresponding parts of congruent triangles are congruent, and the
hypotenuses form a quadrilateral will congruent sides. A rhombus.
With this rhombus four more isosceles triangles can be made.
************************************************
Ethan Castor
castor@netcom.com
Granada High School
Livermore, Ca
************************************************
From: Kevin Scorza & Randy Kuczaj
Grade: 10
School: York High School
1. One way to construct an isosceles triangle is by drawing two
circles overlapping each other forming 2 intersection points. Then
from one of the circles, draw 2 lines from the circle's center point
to the points of intersection of the 2 circles. Next draw a line from
one of the 2 circles' points of intersection to the other point of
intersection, forming an isosceles triangle because all radii are
congruent.
2. A second way of constructing an isosceles triangle is by drawing
a square. Next draw one of the squares' diagonal. This forms two
triangles. Then by erasing 2 corresponding sides of the square that
forms one of the two triangles, you form an isosceles triangle
because on a square, all sides are equal, thus the legs of the triangle
are equal.
3. A third way of constructing an isosceles triangle is by drawing a
rectangle. Then draw the 2 diagonals. This forms 4 equal triangles.
Since the diagonals of a rectangle are perpendicular bisectors of
each other, by erasing the two short sides, one long side, and the
two diagonals beyond their point of intersection, you form an
isosceles triangle.
************************************************
From: Mike Sue, Neil Tucker, Deanne Derego Grade: 10
School: Granada
Answer: One way:
Draw a triangle with 2 congruent adjacent angles. That makes the
opposite legs congruent, which means it can be an isosceles triangle.
Second:
Draw a circle. Draw one chord in the circle but it can't be the
diameter. Next connect the two points with a segment where the
chord meets with the circle and the middle of the circle. This will
give you two radii, and they obviously will both be congruent.
Third: Draw a line segment of any size. Then draw an arc from
one end of the segment and on the other end, another arc that is
congruent to the first. Find where the intersection is and draw line
segments from the segment and the point of intersection. This will
give you an isosceles triangle as well as the others.
[They sent another one at my prompting - see if you can decide
which three are best. - Annie]
In response to your response, draw line segment AB. At each
endpoint draw an arc. Where the arc intersects the segment draw
another so it intersects the previous arc. From each end point draw
a line to its arc intersecting at point C. The segment and these form
an isosceles triangle, triangle ABC.
************************************************
Example 1
Step 1. Construct an angle. Label the vertex C. This angle will be the
vertex angle of your isosceles triangle.
Step 2. Place the pointer of your compass on point C and swing an
arc passing through the two sides of angle C.
Step 3. Label the two points A and B. Construct line AB. You have
constructed isosceles triangle ABC.
Step 4. Use your protractor to measure the base angles of isosceles
triangle ABC.
Example 2
Step 1. Construct a line segment. Label it line AB.
Step 2. Construct an acute angle at point A.
Step 3. Duplicate angle A at point B. Label the new point of
intersection C.
Step 4. Use your compass to compare the sides AB and BC.
Example 3
Step 1. Construct a line segment. Label it line AB.
Step 2. Construct an right angle at point A. Label the other
point C.
Step 3. Connect point B to point C.
Step 4. Use compass to compare sides AB and AC.
************************************************
Dear Annie,
I am sending you a modified version of my solutions
because I have thought of another way of constructing an isosceles
triangle. I have found 6 ways to construct an isosceles triangle.
construction 1.
1. From one point draw 2 straight lines the same length.
2. Connect the ends of the lines, where they are furthest apart,
with a third straight line.
construction 2.
1. From a single point draw two straight lines any angle apart.
2. Place a pair of compasses, set at any length, on the point where
the lines meet and draw an arc which intersects both lines.
3. Draw a third straight line which joins the two points where the
arc intersects the lines.
construction 3.
1. With a pair of compasses draw a large arc of a circle.
2. From the centre of the circle draw 2 lines which meet the arc
at any 2 points.
3. Draw a third line which joins the 2 points lying on the arc.
This method will give you a range of isosceles triangles, each with
2 sides equal to the radius of the arc.
construction 4.
1. Draw a circle and draw in a diameter.
2. At any point on the diameter, draw a line at right angles to the
diameter which intersects the circumference of the circle on each
side of the diameter, at points A & B.
3. Choose one of the 2 points where the diameter intersects the
circle and from it draw 2 lines, one to point A, and one to point B.
construction 5.
1. Draw a line of any length.
2. Set a compass at any length greater than half the length of the
line just drawn.
3. Set the point of the compass at one end of the line and draw an
arc. Do the same at the other end of the line. From the point where
the two arcs intersect draw 2 straight lines, 1 to each end of the
original line.
construction 6. (This is a more practical solution so I'm not sure if
it is what you want.)
1. Take any rectangular piece of paper.
2. Cut it in half along one of its diagonal lines.
3. You now have two right angled triangles. Place the two second
longest sides together along their length(or the 2 short sides together)
so that one triangle is a reflection of the other (i.e., the right angles
are next to each other). Together they form an isosceles triangle.
From Amy Forster, age 11, Home school, Crooked Tree Point,
Cygnet,Tasmania,Australia
Wilkins/Forster family
Crooked Tree Point.Cygnet.
Tasmania. Australia
************************************************
From: Thomas S. Kuo
School: Murray Junior High School, Ridgecrest, California
Grade: 7th
*
* A
* *
* * *
* * *
* * *
* * *
* * *
* * *
* * *
* * *
* * * * * * * * * * *
B * D C
*
Method 1:
(1) Draw a line segment BC.
(2) Expand compass to the length of legs of the isosceles
triangle.
(3) Put needle of the compass at point B and draw a circle.
Put needle of the compass at point C and draw a circle.
These two circles interact at point A.
(4) Connect point A, B, and C. An isosceles triangle with
sides given is built.
[Thomas added some explanation later - do you think it helps?]
If the lengths of two congruent sides and the third side of the
isosceles triangle are given, then the length of BC should be equal
to the length of the third side and the radius of circle should be
equal to the length of the two congruent sides. Then it should be
clear. The radius of circle should be greater than half of the length
of BC or they can not form a triangle.
Method 2:
(1) Draw two line segments AD and BC. They are perpendicular
to each other and interacts at point D.
(2) Make BD = DC = half of length of side of isosceles triangle.
(3) Expand compass to the length of legs of the isosceles triangle.
(4) Put needle of the compass at point B and draw a circle.
The circle and line AD interact at point A.
(5) Connect point A, B, and C. An isosceles triangle with
sides given is built.
[more added later:]
Again, let lengths of two congruent sides and the third side of the
isosceles triangle be given. The steps (2) and (3) above should be
rewritten as follows:
(2) Make BD = DC = half of length of the third side of the triangle.
(3) Expand compass to the length of the two congruent legs of the
triangle.
In method 1, point A is determined by the interaction of two circles.
In method 2, point A is determined by the interaction of one circle
and the bisection line of line segment BC.
Method 3:
(1) Draw a circle centered at point A with radius the length
of legs of the isosceles triangle.
(2) Find any point B on the circle and connect point A and B.
(3) Draw a circle centered at point B with radius the length
of side of the isosceles triangle (other than the equal legs).
This circle will interact with the previous circle at point C.
(4) Connect point A, B, and C. An isosceles triangle with
sides given is built.
[I asked Thomas if he could make this more general]
Let the lengths of two congruent legs and the third leg of the
isosceles triangle be given again. The step (1) and (3) above should
be rewritten as follows:
(1) Draw a circle centered at point A with radius as the length of
the third leg of the triangle.
(3) Draw a circle centered at point B with radius as the length of
the two congruent legs of the triangle. This circle will interact with
the previous circle drawn in (1) at point C.
The difference of this method and the previous methods is that I
draw the congruent leg first.
All methods above are assumed that lengths of legs of the isosceles
triangle are given. However, as stated in method 1, as long as the
length of the congruent legs are greater than half of the length of
the third leg, these methods should work.
************************************************
From: Carolyn Watling
Grade: 9th
School: Granada High School
1. Draw a line AB. Construct the perpendicular bisector of that
line. Label that point C. Connect the points AC and BC with lines.
You now have an isosceles triangle ABC. The two sides, AC and
BC, are equal in length.
2. With a compass, draw a circle of any diameter. Label the center
point "A". Mark any two points on the circle and label them B and
C. Now draw the radius AB and the radius AC. Connect the points
B and C (a chord). You now have an isosceles triangle ABC. The
two sides, AB and AC, are equal in length.
3. Draw any angle ABC. With your compass at vertex B, draw an
arc which intersects both sides, BA and BC, of the angle. Label
these points of intersection D and E and connect them with a
straight line. You now have an isosceles triangle DBE. Sides DB
and BE are equal in length.
************************************************
This is the first time my accelerated geometry class has
participated. There are two eighth graders in the class and twelve
ninth graders. They worked in groups of two. Here are their
solutions with compass and straight edge:
1. Draw a segment. Construct two intersecting arcs with the same
radius, each centered at one of the endpoints of the segment. Draw
segments from the point of intersection to the endpoints of the
segment.
2. Draw a segment. Place compass point on one end of the segment
and make the compass the same size as the segment. Draw a
semicircle. Without changing the compass setting, place the
compass point on the other end of the segment. Draw another
semicircle. Draw segment connecting the two points of intersection
of the two arcs (semicircles). Connect the both endpoints of the
new segment to either of the endpoints of the original segment.
3. Draw a circle. Draw two radii of the circle. Construct the chord
which connects the endpoints of the two radii.
4. Draw a segment. Construct the perpendicular bisector of the
segment (Construct two arcs with congruent radii is greater than
half the segment, one at each endpoint of the segment. Draw a line
through the resulting two intersection points). Connect the original
segment's endpoints to and point along the perpendicular bisector.
5. Draw a point. Put the sharp end of a compass on that point.
Draw an arc of any size. Draw two points anywhere on that arc.
Connect the three points to make an isosceles triangle.
6. Draw a segment. Draw a ray from one endpoint of the segment.
Copy the angle formed by the segment and the ray to the other
endpoint of the segment (Construct an arc across the angle, copy
the arc to the other endpoint, measure the arclength with the
compass, copy that length to the other arc, draw the ray implied by
the endpoint and the intersection of the arcs). The point were the
rays intersect is the third vertex of the triangle (congruent angles
imply congruent sides).
7. Draw an angle. Place the compass point on the vertex and draw
and arc across the angle. Connect the points where the arc and the
sides of the angle intersect.
There are several variations of the same basic concept, but we had
fun trying!
Lisa Lavelle, Teacher, Maret School, Washington, DC
Lauren Austrian Rachel Block Keira Cohen Marie Ghadar
Ann Jackson Philip Kowalczyk Julia Lipton Jenny Lunstead
Peter Miller Justin Schauer Peter Shattuck Lee Teslik
James Whittle Matt Winn
************************************************
Jill Sommer
Mt. St. Joseph Academy
Grade 10
I thought of a few ways to construct an isosceles triangle.
1) Construct a square and one of its diagonals. Then, hide two of
the sides which make a right triangle with the diagonal as its
hypotenuse. At this time, you will have a single right triangle that
happens to be isosceles because the legs of the triangle will have
been 2 sides of the square which are congruent. This would also
apply to a rhombus.
2) Construct a circle and two radii. Then, connect the radii at their
points on the circle, and hide the circle. These steps will yield one
isosceles triangle because radii of the same circle are always
congruent.
3) Construct a segment with its perpendicular bisector. Pick any
point on the bisector and connect it with both endpoints of the
segment. This will be an isosceles triangle because the endpoints of
any segment are equidistant to any point on its perpendicular
bisector.
4) This one was a little difficult for me to visualize. Since the
midpoint of the hypotenuse of a right triangle is equidistant from
each vertex, the segment from the hypotenuse to the right angle and
1/2 of the hypotenuse would be the congruent legs of the isosceles
triangle. The base of the new triangle would be one of the legs of
the right triangle.
************************************************
Michele Weiss
Mount Saint Joseph Academy
There are many ways to construct an isosceles triangle. Here are the
ones I came up with.
1. Construct a circle. Then, construct two radii. Connect the
endpoints of the radii that lie on the circle with a line segment.
Now you have an isosceles triangle.
2. Construct a line segment. Place a compass on each of the
endpoints and construct congruent arcs. Where the arcs meet,
construct a point. Connect that point to each of the endpoints of the
segment. Now you have a second isosceles triangle.
3. Construct a segment. now construct its perpendicular bisector.
Pick a point anywhere on the perpendicular bisector and connect it
to the endpoints of the segment. Now you have a third isosceles
triangle.
4. Construct a square. Construct the one diagonal. You get two
isosceles triangles from one construction!
That's all for now! Bye!
************************************************
Susan Tull & Jackie Mattera
Grade 9
MSJA
There are several possible ways to construct an isosceles triangle.
Here are some of them:
Construct a circle. Construct two radii of the circle. Connect the
points where the lines meet the circle. Since all radii of a circle are
congruent, the triangle is isosceles.
Construct a line segment. Construct the midpoint of the line.
Construct a line perpendicular to the segment at the point. That is
the perpendicular bisector of the segment. Create a point anywhere
on the bisector. Connect that point to the endpoints of the segment.
Since a point on the perpendicular bisector of a segment is
equidistant from the endpoints of the segment, the triangle is
isosceles.
Construct a square. Construct the diagonal of the square. Since all
sides of a square are congruent, the triangle is isosceles.(you could
substitute the square with a rhombus)
*******************************
The following students included sketches as their solutions:
************************************************
Carly Tubbs, grade 9, Fairfield High School
1. Draw a straight line and choose two points on the line. Set the
compass to one setting. Place the point of the compass on one point
and draw an arc. Using the same setting, place the point of the
compass on the other point and draw an arc which intercepts the
first arc. Draw a point where they intersect. From that point, draw
a line to 1 point on the line, then draw another line to the other
point.
2. Draw a point. Put the point of the compass on the point and
draw a line, with a dot on the end. Put the point of the compass on
one point, and the pencil on the other. Using that setting, keeping
the point on the point, draw an arc. Connect the two bottom points.
3. Draw an angle on the paper. Put the compass at the vertex and
draw an arc that intersects both sides of the angle. On the other
point on the base line, draw another arc, the same setting. On the
arc intercepting the angle, put the point of the compass on one
point of the arc, and the pencil point on the other. Keeping that
setting, put the pencil point on the other arc and draw an arc
intercepting that arc and a point. Draw a line through the point
connecting to the base line.
4. Draw an angle and putting the point of the compass on the
vertex, draw an arc through the angle. From each point on the arc
that intercepted the angle, draw two other arcs so that they
intercept each other. Connect the vertex and the point together.
Then draw a line through the arc marks, through the angle sides.
5. Draw a circle. Draw a point outside the circle. From that point,
construct two tangent lines to the circle. Where they meet at the
circle, connect the two points.
************************************************
College Park High School, Pleasant Hill, California
Philip Horner
1. The first way I know of to make an isosceles triangle is to draw
a circle. Keeping the compass at the same measurement, pick a
point on the circle and place the compass axis on it. Then swing the
compass so it intersects with the circle twice. Connect these three
points so each is connected with the other two. You now have an
isosceles triangle.
2. Pick a point off of a straight line. Place the compass on this
point. Swing the compass so that it intersects the line at two
different points. Connect these three points with a straight edge so
that each point is connected to the other two. You now have an
isosceles triangle.
3. Pick a point an place your compass on it. Swing the compass so
that it makes an arc. Connect the endpoints on the arc to the
original point where you placed your compass. Connect the end
points to each other with a straight edge. You now have an isosceles
triangle.
4. Draw a circle on a coordinate plane, using the point (0,0) as the
center point. Then connect the x intercept to the y intercept and the
x intercept and then connect the y intercept to the x intercept and
you have and isosceles triangle.
************************************************
Asa Sharma
1. Draw a line. Construct a line perpendicular to this line and
connect at end points. Use a compass to make an arc from A.
Connect where they intersect.
2. Draw a line. Pick a point A. Use a compass to make a circle
through line. Draw a line to perpendicular to the line through A.
Connect points B to C, B to D.
3. Draw an angle ABC where BC is the longest line. Then construct
a line perpendicular to BC through angle, intersecting BC at D.
ABC is an isosceles angle.
************************************************
Angie Bush
One way you can construct an isosceles triangle is to draw a line
segment and construct a perpendicular bisector to it. By drawing
diagonal lines from the top of the perpendicular bisector to the
endpoints of the segments you get a triangle with two equal sides.
Another is to draw a circle with the center on the origin of the x, y
axis. Connect the two x intercepts with one of the two y intercepts.
Another way is to swing an arc from the origin of an x,y axis that
intersects both the x and the y. Then connect to origin to the place
that intersects the x and the y and connector the points on x and y.
************************************************
Shastina Larson and Rebecca Pearson have similar answers.
************************************************
Problem of the Week, April 15-19
Give at least three different ways to construct a isosceles triangle.
1. Draw a segment of any length. Label the segment AB. From
point A, use a compass to construct an arc with length greater than
half the length of AB. Construct a congruent arc from point B that
intersects the first arc. Label the point of intersection of the two
arcs C. Triangle ABC is an isosceles with congruent sides AC and
BC.
2. From any point A, construct an arc of any length. Mark any two
points on the arc as points B and C. Use a straightedge to make
segments AB, AC, and BC. Triangle ABC is an isosceles triangle
with congruent sides AB and AC.
3. Draw a segment of any length. Label the segment AB. Construct
the perpendicular bisector of segment AB and label it as line m.
Select any point on line m and label it C. Draw segments AC and
BC. Triangle ABC is an isosceles triangle with congruent sides AC
and BC.
4. Use a straightedge to draw any non-straight angle. Label the
vertex A.
From vertex A, use a compass to construct an arc that intersects
both sides of the angle. Label the intersections as B and C. Draw
segment BC. Triangle ABC is an isosceles triangle with congruent
sides AC and AB.
5. Draw a segment of any length. Label the segment DB. Construct
the perpendicular bisector of segment DB and label it as line m.
Select any point on line m and label it C. Label the point where line
m intersects segment DB as A. Triangle ABC is a right triangle.
Construct the perpendicular bisector of hypotenuse BC to find its
midpoint. Label the midpoint as E. E is equidistant from all three
vertices of the triangle. Therefore, triangles ECA and EAB are
both isosceles triangles.
6. Construct a circle and label the center as O. Mark an exterior
point to the circle as point P. Draw segment OP. Construct the
perpendicular bisector of OP to find it's center. Label the center as
M. With compass length of MO or MP, construct a circle with
radius MO or MP and center M. Label the points where circle #2
intersects circle #1 as points X and Y. Draw segments PX, PY, and
XY. Triangle PXY is an isosceles triangle with congruent sides PX
and PY (segments formed by constructing tangents to a circle from
an external point to the points of tangency are congruent).
Respectfully submitted,
Gary Boraas
Geometry Instructor
Lac qui Parle Valley High School
The Math Forum is a research and educational enterprise of the Drexel School of Education.