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Triangles
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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
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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 9-12
Access to these problems requires a Membership.
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Another Ambiguous Angle?
- Annie Fetter
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Geometry, difficulty level 3. Given isosceles triangle ABC, with BE perpendicular to AD. If angle ACB is x, what is angle CBE?
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The Apprehendable Angle
- Annie Fetter
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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?
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Building a Bridge Across a Chasm
- Annie Fetter
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Geometry, difficulty level 2. So they're standing on the edge of this chasm. It isn't really that far across, but they need to find a log or tie together enough rope to reach a tree on the other side. So they mark a spot on the edge of the chasm directly across from the tree. Then they walk 20 paces along the chasm, and she stays there. He walks another 20 paces and marks the spot. Then he turns ninety degrees and walks away from the chasm until she's directly in the line of sight between him and the tree on the other side. Then he claims that the distance he just walked from the edge of the chasm is the same as the distance across the chasm.
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Building a Dog Enclosure
- Annie Fetter
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Geometry, difficulty level 3. Two people are arguing about whether three sections of fencing 6, 8, and 10 feet long will enclose more or less area than three sections 6, 8, and 12 feet long.
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The Center of Gravity of a Quadrilateral
- Annie Fetter
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Geometry, difficulty level 3. Find the center of gravity of a quadrilateral.
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Chords and Arcs
- Annie Fetter
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Geometry, difficulty level 3. A chord of a circle is the hypotenuse of an isosceles right triangle whose legs are radii of the circle. The radius of the circle is 6 times the square root of 2. What is the length of the minor arc subtended by the chord?
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Circle, Hexagon, & Triangle
- Annie Fetter
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Geometry, difficulty level 2. A regular hexagon and an equilateral triangle share three vertices and are inscribed in a circle with a radius of 8 units. What is the area of the region between the two polygons?
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A Circle Inscribed in an Isosceles Triangle
- Ilmar Vitsut
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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
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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).
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Coordinated Triangle?
- Annie Fetter
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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
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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 units^2, where can the third vertex be? What if the triangle is also isosceles?
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Drawing Out a Decagon
- Annie Fetter
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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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Extending the Enneagon
- Annie Fetter
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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?
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Finding Formulas for Medians from Edgelengths
- Annie Fetter
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Geometry, difficulty level 3. Write a formula for finding the median of a triangle in terms of the three sides of the triangle.
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Find This Angle
- Annie Fetter
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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?
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Fix This Picture
- Annie Fetter
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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
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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
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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?
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Integer Triangle Angles
- Annie Fetter
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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?
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Mad About Medians?
- Annie Fetter
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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
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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
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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?
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Making Triangles I
- Annie Fetter
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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
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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?
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Napoleon's River
- Annie Fetter
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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
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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?
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Proving PD
- Annie Fetter
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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.
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Quad Query
- Annie Fetter
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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?
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Returning to Napoleon's River
- Annie Fetter
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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?
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Slicing a Cube
- Annie Fetter
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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.
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Splitting up a Triangle
- Annie Fetter
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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?
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The North Hunterdon Triangle
- Annie Fetter
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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.
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Triangles and String
- Annie Fetter and Steve Risberg
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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
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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
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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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What's My Angle?
- Annie Fetter
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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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