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Triangles

Problems in this section may deal with the basic definitions of
medians, altitudes, angle bisectors, and perpendicular bisectors
or the defining characteristics of the different types of triangles.
Algebraic problems focus on the sum of the angles in a triangle.
Most of the problems should be appropriate for the beginning
geometry student.
Problems pertaining to the
Pythagorean Theorem
or Special Right
Triangles can be found in their own categories.
 Related Resources

Interactive resources from our Math Tools project:
Geometry: Triangles
The closest match in our Ask Dr. Math archives:
High School: Triangles
NCTM Standards:
Geometry Standard for Grades 912
Access to these problems requires a Membership.

Another Ambiguous Angle?
 Annie Fetter

Geometry, difficulty level 3. Given isosceles triangle ABC, with BE perpendicular to AD. If angle ACB is x, what is angle CBE?
... more>>

The Apprehendable Angle
 Annie Fetter

Geometry, difficulty level 2. Triangle ABC is isosceles with a vertex angle (at the top) of 50 degrees. If D is on AC, E is on BC, and AD = BE, what is the measure of angle ADE?
... more>>

A Circle Inscribed in an Isosceles Triangle
 Ilmar Vitsut

Geometry, difficulty level 4. Find the radius of a circle inscribed in an isosceles triangle with sides 12, 12, and 8.
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Construct an Isosceles Triangle
 Annie Fetter

Geometry, difficulty level 2. Give at least three different ways to construct an isosceles triangle (a construction can be repeated over and over and in this case will always yield an isosceles triangle).
... more>>

Coordinated Triangle?
 Annie Fetter

Geometry, difficulty level 3. Triangle ABC has an area of 24 units^2. Point A is at (6,0); point B is at (10,0). Where can point C be? What if the triangle is
isosceles?
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Descartes' Triangle?
 Annie Fetter

Geometry, difficulty level 3. Two vertices of a triangle are located at (0,6) and (0,12). If the area of the triangle is 12 square units, where can the third vertex be? What if the triangle is also isosceles?
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Drawing Out a Decagon
 Annie Fetter

Geometry, difficulty level 2. Extend sides AB and IH of the regular decagon ABCDEFGHIJ until they intersect. What is the measure of the acute angle at this intersection?
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Eight Sticks and Zero Triangles
 Annie Fetter

Geometry, difficulty level 3. Help Imeir and Dat answer this question about sticks of integer
length that cannot make triangles.
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Extending the Enneagon
 Annie Fetter

Geometry, difficulty level 3. Extend the sides AB and ED of the regular enneagon ABCDEFGHI until they intersect. What is the measure of the acute angle at this
intersection? Extend sides AB and EF. Now what's the measure of that acute angle at the intersection?
... more>>

Find This Angle
 Annie Fetter

Geometry, difficulty level 3. Given triangle ABC, where E is on AB and DB is perpendicular to AE. AC = BC, angle ACB is twice the measure of angle CBD, and angle DBE = 90 degrees. What is angle CBD?
... more>>

Fix This Picture
 Annie Fetter

Geometry, difficulty level 2. Figure out what's wrong with this picture that includes a circle and a rhombus.
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The Folding Ruler
 Annie Fetter

Geometry, difficulty level 2. Describe all of the possible triangles that can be made with the full length of a 6 foot long folding wooden ruler.
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Framed!
 Annie Fetter

Geometry, difficulty level 3. Given a triangular painting with dimensions 20 x 14 x 10, and three different ways you could frame it, which frame will take the least amount of material?
... more>>

How Long is the Base?
 Annie Fetter

Geometry, difficulty level 3. Find the length of the base of a particular isosceles triangle.
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Integer Triangle Angles
 Annie Fetter

Geometry, difficulty level 1. The degree measures of a triangle are x, y, and z, and x, y, and z are
all integers where x < y < z. What's the largest possible value of z?
What's the smallest?
... more>>

Mad About Medians?
 Annie Fetter

Geometry, difficulty level 3. The medians drawn from the vertices of the acute angles of a right triangle are 5 and sq root of 40. What is the length of the hypotenuse?
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Making a Map
 Annie Fetter

Geometry, difficulty level 3. Given a list of relationships between five different objects, explain which ones you could uniquely place on a map.
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Making Scalene Triangles
 Annie Fetter

Geometry, difficulty level 2. A scalene triangle has sides of 8, x, and 19. The sides are listed in order from smallest to largest. What are all of the possible integer values for x?
... more>>

Making Triangles I
 Annie Fetter

Geometry, difficulty level 2. Take a scalene triangle. The lengths of the sides, in size order, are 5, x, and 15. What are the possible integral values for x?
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Making Triangles II
 Annie Fetter

Geometry, difficulty level 2. Given a scalene triangle with sides, listed from shortest to longest, of 6, x, and 16, what are all the possible integer values for x?
... more>>

Napoleon's River
 Annie Fetter

Geometry, difficulty level 2. A soldier uses the visor of his cap to measure the distance across the river.
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The Protean Perimeter
 Annie Fetter

Geometry, difficulty level 3. Given isosceles triangle ABC, with AC=AB=5 inches and angle CAB greater than or equal to 60, what's the perimeter of the triangle if all the edgelengths must be integers?
... more>>

Proving PD
 Annie Fetter

Geometry, difficulty level 3. AOD is a diameter of the circle with center O. B is any point on the
circle that isn't A or D. A tangent is drawn to the circle at point B.
A line is drawn through O parallel to AB, meeting the tangent at P.
Prove that PD is a tangent to the circle.
... more>>

Quad Query
 Annie Fetter

Geometry, difficulty level 2. Given quadrilateral ABCD, with E the midpoint of AB and F the midpoint of AD. The area of FAEC is 13. What's the area of ABCD?
... more>>

Returning to Napoleon's River
 Annie Fetter

Geometry, difficulty level 3. A soldier stands on the bank of a river and uses his cap to measure its width. How far off will he be if he drops his head even one degree?
... more>>

The Shortest Possible Side
 Annie Fetter

Geometry, difficulty level 2. Find the shortest possible side length of a triangle with a given
perimeter.
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Slicing a Cube
 Annie Fetter

Geometry, difficulty level 3. Given a cube with a surface area of 54 cm^2, find the perimeter of the square cross section and that of the largest rectangular cross section of that cube.
... more>>

Splitting up a Triangle
 Annie Fetter

Geometry, difficulty level 4. Take any triangle ABC. Construct D and E as the midpoints of BC and
AB, respectively. Now construct DF and EF, where F is any point on AC.
How are the areas of the triangles related to the area of the
quadrilateral?
... more>>

The North Hunterdon Triangle
 Annie Fetter

Geometry, difficulty level 2. Given points O:(0,0) and Q:(a,0), describe the possible locations of
point P such that triangle OPQ is first isosceles and then equilateral.
... more>>

Tied in Knots
 Annie Fetter

Geometry, difficulty level 2. How many triangles can be made from the knotted rope?
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Triangles and String
 Annie Fetter and Steve Risberg

Geometry, difficulty level 2. Given a loop of string 12" long, find all possible triangles of integer side lengths that it can form, and explain whether they are scalene, isosceles, or equilateral, as well as acute, right, or obtuse.
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Triple Tango
 Annie Fetter

Geometry, difficulty level 1. Sort six triangles into three pairs so that the two triangles in each pair can be put together to form a larger triangle without overlapping.
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Visiting Friends
 Annie Fetter

Geometry, difficulty level 2. Annie and Suzanne live near the same train station. If you are told
how long it takes to walk from each of their houses to the train
station, what can we say about how long it takes to walk at this rate
directly from Annie's to Suzanne's?
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Voulez Vous des Voussoirs?
 Annie Fetter

Geometry, difficulty level 2. Voussoirs are trapezoidal chunks of stone used to build stone arches.
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What's My Angle?
 Annie Fetter

Geometry, difficulty level 2. Given line AE, point C between A and E, ray CB perpendicular to line AE, and point D on the same side of AE as point B. If CD equals DE and angle CDE is x degrees, what is the angle measure of angle BCD?
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Where's J?
 Annie Fetter

Geometry, difficulty level 2. Given an isosceles triangle with one base vertex (O) on the origin, the other base vertex (J) on the positive xaxis, and the top vertex at the point (e,f), find the coordinates of J.
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