InterMath Dictionary

Triangles

 Altitude of a Triangle  Hypotenuse  Pythagorean Theorem
 Angle-Angle-Side (AAS)     Pythagorean Triple
 Angle-Side-Angle(ASA) Incenter  
 Area  Leg  Side-Angle-Side (SAS)
 Centroid  Median of a Triangle  Side-Side-Side (SSS)
Circumcenter    Triangle
 Congruent Triangles  Orthocenter  Triangle Inequality
  Corresponding Angles  Perpendicular Bisector  Triangular Number
 Corresponding Sides    


Triangles
Triangle - Triangle (3-angles) is a shape (polygon) with 3 straight sides. Three is the smallest number of sides needed to make a closed figure.

Naming of Triangles:


Classification of Angle by Location:

Interior/Exterior:

The angles in the triangle are refered to as interior(internal) angles (the blue, yellow and green angles). Triangles also have exterior(external) angles (the red angles). Extending any side of the triangle either direction creates an exterior angle. An exterior angle and its adjacent interior angle are supplementary; they make up a straight line which is 180.

 


Uses of Triangles in the Real World

Triangles are rigid strong structures used for rafters in building and curved domes. Engineers use the shape for bridges. Look around to see triangles in your surroundings. The Egyptians used triangles in the pyramids.

These shapes are used to find the distance of a given point to two other points which are a known distance apart. This is known as triangulation and can be used for measuring around corners or digging a tunnel.

Can you find other places where triangles are used?

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Parts of a Triangle

Hypotenuse - In a right triangle, the side opposite the right angle.
Leg of a Triangle - Either of the two shorter sides of a right triangle. These two sides together form the right angle in the right triangle.
Median of a Triangle - A segment is a median of a triangle if and only if its endpoints are a vertex of the triangle and the midpoint of the side opposite the vertex.
Perpendicular Bisector- A perpendicular line that passes through the midpoint of a segment.

The hypotenuse is the side of a right triangle that is directly across from the right angle (indicated by the black square in the picture at the left).

 

 

 

 

The shorter sides of a right triangle are called the legs of the triangle (sides AB and BC in the picture at the right). Together the legs form the right angle of the right triangle.

 

 

 


The medians of a triangle are line segments formed by connecting a vertex to the midpoint of the opposite side. Since there are three vertices and three sides, there are three medians. Medians can be formed in all types of triangles, not just right triangles.

 

Does the median always fall inside the triangle? If so, why do you think so. If not, why do you think not?

In any triangle, construct all three medians. Do they all meet at a single point? (See Centers of Triangles)


A perpendicular bisector is a perpendicular line that passes through the midpoint of a line segment, like a side of a triangle. In the triangles below, the lines are perpendicular (indicated by the black square) at the midpoint (indicated by points and ) of a side of the triangle.

 

In any triangle, construct all three perpendicular bisectors. Do they all meet at a single point? (see Centers of a Triangle)

Both the perpendicular bisector and the median of a triangle pass through the midpoints of the sides of a triangle. Under what circumstances will the perpendicular bisector and the median actually coincide?

Check for Understanding

 In the interactive diagram at the right, identify a: Sorry, this page requires a Java-compatible web browser.

 Enter each answer in the left column followed by pressing tab

Graded Response
  a) median  a)  
 b) leg  b)  
  c) hypotenuse  c)  
  d) perpendicular bisector  d)  

External Links to Extend Basic Understanding

Harcourt's Interactive Math Glossary: Read another defintion of leg, hypotenuse, and perpendicular bisector.

Medians and Perpendicular Bisectors at ThinkQuest.org: Read another definition of median and perpendicular bisector.

Related Investigations on InterMath to Challenge Teachers

Find the Hidden Treasure: Determine a shortcut to find a treasure inside of a triangle.
Balancing the Centroid
: Describe how a triangle's median is divided.
Medians of a Triangle: Determine the relationship between the coordinates of the vertices of a triangle and its centroid.
Capture the Flag
: Locate the best starting position in a game of capture the flag.
Perpendicular Bisector
s: Describe the significance of the perpendicular bisectors of two segments that have endpoints on a given circle.


Altitude of a Triangle - The perpendicular distance between the apex and the base of the triangle. Also called the height of a triangle.
Height of a Triangle- The perpendicular distance between the apex and the base of a triangle. Also called the altitude of a triangle.

Examples of Altitudes

 

The altitude (h) of this triangle is shown in red.

You can find the altitude or the height of a figure by measuring the segment (h) that is perpendicular to the base and that meets the opposite vertex (E). This segment h can also be called the altitude.

 

 

Sometimes the altitude will fall outside the figure, as in the triangle to the right. In this case, you need to extend the base of the figure so that it intersects the altitude at a right angle.

 

 

 


Non-Example of Altitude

 

In this picture, the green segment is not an altitude of the triangle since it is not perpedicular to a base (BC) of the triangle. The altitude is indicated in red.

 

 

 

 

Check for Understanding

 In the interactive diagram at the right, identify: Sorry, this page requires a Java-compatible web browser.

 Enter each answer in the left column followed by pressing tab

Graded Response
  a) an altitude if the base is BC  a)  
 b) a base if the altitude is BD  b)  

External Links to Extend Basic Understanding

Altitudes at ThinkQuest.org: Read another defintion of altitude.

Related Investigations on InterMath to Challenge Teachers

Altitude to the Hypotenuse: Investigate the relationship between sides of right triangles when an altitude is drawn to the hypotenuse of a right triangle.
In or Out?: Determine the location of the altitudes in a triangle.
Area formulas
: Investigate the area of triangles in a variety of ways.

 

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Area of a Triangle

Area - The measure of the region formed by the triangle and its interior. It is calculated by using the formula where b is the length of the base of the triangle and h is its height.

 

Where do we get the formula ? Consider the following argument that uses the area of a parallelogram.

Suppose you have a triangle with height h and base b. Reflect or flip a copy of the triangle to form the parallelogram on the right. Given that the area of the parallelogram is bh and since the original triangle forms only half of the parallelogram, the area of the triangle is .


Consider another argument that relates the area of a rectangle to the area of a triangle.

 

          

 (a)

 (b)

 (c)

 (d)

Fold a given triangle so that you divide its height in half (along the dotted line in (a)). We obtain the trapezoid pictured in (b). Now we fold along the dotted lines in (c) to obtain the rectangle in (d). We know the height of this rectangle is half the height of the triangle because we folded it in half. But why is the base of the rectangle half that of the triangle? The area of the rectangle is 1/4 bh, but we need 2 of these rectangles to make the original triangle (why?) so that when we multiply 1/4 bh by 2, we get 1/2 bh, the area of the original triangle.

Check for Understanding

Determine the area of triangle ABC based on the lengths given in the diagrams.

 Enter each answer in the left column followed by pressing tab

Graded Response
a)  a)cm2  
b)  b)cm2  
c)  c)cm2  

External Links to Extend Basic Understanding

Area of Triangles at AAAMath.com: See a lesson, formulas, and practice problems about the area of triangles.

Related Investigations on InterMath to Challenge Teachers

Area formulas: Investigate the area of triangles in a variety of ways.
Triangle Inside a Rectangle
: Examine the area and perimeter of a triangle compared to a rectangle.
Optimal Triangles
: Determine the largest area of a triangle given a restricted perimeter.
Rationalize this!
: Find the a right triangle with rational side lengths, and a hypotenuse numerically equal to the area of the triangle.
Triangles in a Trapezoid
: Determine the relationship between the areas of triangles formed by the diagonals of a trapezoid.


Square Units?

Area is in square units since you are multiplying one length by another length. For instance, if the base of a triangle is 12 inches and the height is 9 inches, then the area of the triangle can be found by multiplying


Congruency

Corresponding Angles - Angles that have the same relative positions in geometric figures.
Corresponding Sides - Sides that have the same relative positions in geometric figures.
Congruent Triangles -
Triangles that have the same size and shape.
Angle-Angle-Side (AAS)
- A congruence property that states if two angles and the non-included side of a triangle are congruent, respectively, to two angles and the non-included side of another triangle, then the two triangles are congruent.
Angle-Side-Angle (ASA)
- A congruence property that states if two angles and the included side of a triangle are congruent, respectively, to two angles and the included side of another triangle, then the two triangles are congruent.
Side-Angle-Side (SAS) - A congruence property that states if two sides and the included angle of a triangle are congruent, respectively, to two sides and the included angle in another triangle, then the two triangles are congruent.
Side-Side-Side (SSS) -
A congruence property that states if three sides of a triangle are congruent, respectively, to the three sides of another triangle, then the two triangles are congruent.

Congruent Triangles

Congruent triangles have the same size and shape. In order for triangles to be congruent, corresponding parts must be congruent. Corresponding parts mean that the parts (angles or sides) are in the same relative position in each of the triangles. While congruent triangles have the same size and shape, they may not appear to be congruent because they may be oriented on the page or in space differently. So you need to find those corresponding parts and check them out!

The following pairs of triangles are congruent. We use particular marks to indicate which parts of triangles are congruent to each other. Congruent sides have the same number of hash marks. For example, in Pair A, sides AB and DE have one hash mark indicating they are congruent. Also in Pair A, sides AC and DF have 3 hash marks, indicting congruency. Congruent angles are identified by the number of arcs across the angles. For example, in Pair B, angles A and A' have 3 arcs, indicating congruency. In Pair C, angles B and B' are congruent, indicated the two arcs across both angles.

 

 Pair A

 

  Pair B

 

  Pair C

 

Angle-Side-Angle (ASA)

According to the ASA congruence property, to show congruency of two triangles, it is sufficient to verify that two angles and the included side of one triangle are congruent respectively to the corresponding parts of another triangle.

Angle-Angle-Side (AAS)

According to the AAS congruence property, to show congruency of two triangles, it is sufficient to verify that two angles and the non-included side of one triangle are congruent respectively to the corresponding parts of another triangle.

Side-Angle-Side (SAS)

According to the SAS congruence property, to show congruency of two triangles, it is sufficient to verify that two sides and the included angle of one triangle are congruent respectively to the corresponding parts of another triangle.

The following picture shows two congruent triangles according to the SAS congruence property. In the picture, the triangle ABC is isosceles (Sides AB and AC are congruent). If we construct the angle bisector for A (indicated by P), we get two congruent angles, angle CAP and angle BAP. The segment AP is congruent with itself. Thus, we have two sides and the included angle in one triangle being congruent with the corresponding parts of the other triangle. So, by SAS triangles CAP and BAP are conrguent.

Side-Side-Side (SSS)

According to the SSS congruence property, to show congruency of two triangles, it is sufficient to verify that the three sides of one triangle are congruent respectively to the corresponding sides of another triangle.

Did You Know?

In his books Elements, Euclid showed that for SAS, ASA, AAS and SSS, if three corresponding parts of two triangles are congruent, then all six parts have corresponding congruent parts. However, there are instances in which five of the six parts of one triangle are congruent to five of the six parts of another triangle without the triangles being congruent! Consider the following pair of triangles. Five parts of one triangle are congruent to five parts of the second triangle. But why are they not congruent?

Check for Understanding

 Determine the property that would verify triangle ABC congruent to triangle FED based on the hash marks and arcs on the diagrams. Choose each answer from the menus  
  a)    a)  
  b)    b)  
  c)    c)  
  d)    d)  

External Links to Extend Basic Understanding

Harcourt's Interactive Math Glossary: Read another definition of congruent, congruent figures, ASA, SAS, and SSS.


Pythagorean Theorem

Pythagorean Theorem - In a right triangle, the square of the length of the hypotenuse equals the sum of the squares of the lengths of the legs.

This theorem is one of the most famous and useful theorems in geometry. Pythagoras (580-496 B.C.), a Greek mathematician, was the first to prove this theorem, so it has become known as the Pythagorean Theorem. However, it appears that the Babylonians may have been aware of the theorem because they were using its converse (If the square of the length of the hypotenuse equals the sum of the squares of the lengths of the legs, then the triangle is a right triangle) more than 1300 years before Pythagoras.

It is a theorem that states a relationship that exists in any right triangle. If the lengths of the legs in the right triangle are a and b and the length of the hypotenuse is c, we can write the theorem as the following equation:

 

A model of the theorem is shown below. Squares have been constructed on the sides of the right triangle and the area of each square has been dissected into squares. Count the gridded cells in the two squares constructed on the legs of the triangle. They should add up to the number of gridded cells in the square constructed on the hypotenuse.


Many people have come up with their own version of the proof of the Pythagorean Theorem. Even President Garfield wrote a special version of the proof in 1876. Click to see Garfield's proof of the Pythagorean Theorem.

Check for Understanding

Find the lengths of the missing sides on the right triangles. Round your answer to the nearest tenth of a centimeter.

 Enter each answer from the left column followed by pressing tab

Graded Response
a)  a)cm  
b)  b)cm  
c)  c)cm  

External Links to Extend Basic Understanding

Harcourt's Interactive Math Glossary: Read another definition of the Pythagorean Theorem.

Pythagorean Theorem at Math.com: See a lesson about the Pythagorean Theorem, and test yourself with the included questions.

Pythagorean Theorem at Cape Canaveral: Read about Pythagoras and see an animated explanation of the theorem.

Linda's website: Find a list of various links to information about the Pythagorean Theorem.

Related Investigations on InterMath to Challenge Teachers

Interactive Pythagoras: Use the concept of area to "prove" the Pythagorean Theorem
Squaring with Squares
: Draw area connections to the Pythagorean Theorem.
Difference of Squares
: Find positive integers which can be written as sum or difference of squares.
As the Crow Travels
: Can you devise a strategy that you travel in a car exactly twice the distance as a crow flies, but end up at the same spot?


Pythagorean Triple

Pythagorean Triple - A set of three non-zero whole numbers, a, b, and c, such that .

There are infinitely many Pythagorean triples. Some examples are: (3, 4, 5); (11, 60, 61); (5, 12, 13).

Jackie thinks she has discovered a way to generate Pythagorean triples. Here is her method:

Take any odd number. (Let's use 17.) Square it. (You get 289.) Subtract 1 and divide by 2. (This produces 144.) Add 1. (The result is 145.) Now use these numbers to form a Pythagorean triple:

Does Jackie's method always work? Try another example. Can you prove that her method always works, or find a counter-example when it doesn't?

Check for Understanding

Answer the following question about Pythagorean Triples.

 Enter each answer followed by pressing tab

Graded Response
Determine a Pythagorean Triple with lowest value 8.  (8, , )  

External Links to Extend Basic Understanding

Pythagorean Triples at Neat Math: Read another short definition of Pythagorean Triples, and see a method to generate them.


Triangle Inequality

Triangle Inequality - The sum of the lengths of two sides of a triangle is always greater than the length of the third side.

According to Triangle Inequality, there are certain sets of measurements that cannot be used to form triangles. For example, it is impossible to form a triangle with sides measuring 2, 3, and 6 units (2 + 3 < 6).

 

The animation below displays how if given a line segment of three inches, a triangle is not formed until HG + IE are greater than three inches. Notice that when HG + IE equal three inches, they form the line segment HI. This special case is called a degenerate triangle, or in other words no triangle.

 

 

If you know the measures of two sides of a triangle, can you use the Triangle Inequality to determine the greatest and least possible measure of the third side of the triangle?

Check for Understanding

Determine if the following segment lengths form a triangle. Choose each answer from the menus  
  a)  {4, 5, 8}  a)  
  b)  {6, 4, 2}  b)  
  c)  {-2, -2, -2}  c)  

External Links to Extend Basic Understanding

Triangle Inequality at ThinkQuest.org: Read another explanation of triangle inequality.

Related Investigations on InterMath to Challenge Teachers

To Be or Not to Be: Determine the necessary conditions to create a triangle.
Forming a Triangle
: Given certain circumstances, determine the probability of forming various triangles.
The Third Side
: What are the possible perimeters that a given triangle can have?
Prime Scalene
: Find the smallest possible perimeter of a scalene triangle.


Centers of a Triangle
Circumcenter - The center of the circle circumscribed around a given triangle. Also, the point of intersection of the perpendicular bisectors of the sides of a given triangle.
Centroid -
The point of intersection of the medians of a triangle. Also, the point in a figure or solid which is the balance point, or center of gravity.
Incenter - The center of a circle inscribed in a given triangle. Also, the point of intersection of the angle bisectors of a triangle.
Orthocenter -
The point of intersection of the altitudes of a triangle.

 

The perpendicular bisectors of the sides of a triangle intersect at a point called the circumcenter of a triangle. This same point is the center of the circumscribed circle around the triangle.


 

The medians of a triangle intersect at a point called the centroid. If you were to cut out this triangle and try to balance it on a pencil point, the balance point is the same as the centroid.

Can the centroid ever lie outside the triangle? Why or why not?

 

 


 

 

The angle bisectors intersect at a point called the incenter.

 

 

 

Can the incenter ever lie outside the triangle? Why or why not?


 

 

The altitudes of a triangle intersect at a point called the orthocenter.

 

What does "ortho" mean? Why do you think this point is called the "orthocenter"?

 

 

Can the orthocenter ever lie outside the triangle? Why or why not?

 

Check for Understanding

Determine if the following diagrams form a circumcenter, centroid, incenter, or orthocenter.

 Enter each answer in the left column followed by pressing tab

Graded Response
  a)  a)  
 b)  b)  
  c)  c)  
  d)  d)  

Related Investigations on InterMath to Challenge Teachers

Balancing the Centroid: Describe how a triangle's median is divided.
Medians of a Triangle: Determine the relationship between the coordinates of the vertices of a triangle and its centroid.
Inscribed in a Triangle
: Construct a circle inscribed in a triangle so that it will always remain inscribed in the triangle.
Moving Walls
: Determine the ideal position to stand when you are surrounded.
Balancing the Triangular Totter
: Locate the center of mass of a triangle.


Triangular Numbers
Triangular Numbers - The numbers 1, 3, 6, 10, ... are called triangular numbers because the number of dots representing the number can be arranged into a triangle.

Consider the pattern formed by these dots.

The number used to describe each "triangle" is called a triangular number. Given a number, can you arrange that many dots into a triangle? If you can, you have identified a triangular number. The sequence of triangular numbers starts with 1, 3, 6, and so on.

Check for Understanding

Answer the following question about triangular numbers.

 Enter each answer followed by pressing tab

Graded Response
What are the two terms that follow 1, 3, and 6 in the triangular number sequence?   and  

External Links to Extend Basic Understanding

Harcourt's Interactive Math Glossary: Read another definition of triangular numbers.


 

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