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triangles

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by

Şeyma Erbaş

on 14 November 2014

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Transcript of triangles

What is
triangle?

What is fractal?

Area of a triangle

First reminded triangle

Triangle:Triangles are polygons with the least possible number of sides (three).

Classifying Triangles by angles

Bisector: A line that splits an angle into two equal angles.
("Bisect" means to divide into two equal parts.)
Pythagorean Theorem

Area of a triangle

Scalene triangle

Isosceles triangles

Equilateral triangle

Classifying Triangles by Sides

A triangle consists of three sides and three vertices.

A Brief History:
Ahmes' papyrus is the first written source mentioned about triangles (in BC 1550 ,Egypt)
What is the formula for finding the
area of a triangle?
How is it related to the formula for
finding the area of a rectangle?
TRIANGLE
Problem 51 of Ahmes (Rhind) Papyrus gives the area of an
isosceles triangle as half of the base times the height,
justiﬁed by moving half of the isosceles triangle and
forming a rectangle.
VERTEX is a corner of the triangle. Every triangle has three vertices.

BASE of a triangle can be any of the three sides, usually the one drawn at the bottom. You can pick any side you like to be the base. Commonly used as a reference side for calculating the area of the triangle.

ALTİTUDE of a triangle is the perpendicular from the base to the opposite vertex. Since there are three possible bases, there are also three possible altitudes.
MEDIAN of a triangle is a line from a vertex to the midpoint of the opposite side.

INTERIOR ANGLE is three angles on the inside of the triangle at each vertex.

EXTERIOR ANGLE is the angle between a side of a triangle and the extension of an adjacent side.
ACUTE TRIANGLE is a triangle having all acute angles (less than 90 degree) in its interior

OBTUSE TRIANGLE is a triangle having an obtuse angle (greater than 90 degree but less than 180 degree) in its interior.
RIGHT TRIANGLE is a triangle that has a right angle in its interior.

EQUIANGULAR TRIANGLE is a triangle having all angles of equal measure.
EQUILATERAL TRIANGLE is a triangle with all three sides equal in measure

ISOSCELES TRIANGLE is a triangle in which at least two sides have equal measure

SCALENE TRIANGLE is a triangle with all three sides have equal measures

A fractal is a never-ending pattern.
Fractals are infinitely complex patterns that are self-similar
across different scales. They are created by repeating a simple process over and over in an ongoing feedback loop.

PYTHAGORAS
Pythagoras asserted Pythagorean theorem. It is the most known and important theorem of him. He is known as the father of numbers. He was born in Sisam island (BC 580-BC 500). He lived in the south part of Italy and found a school there. He was also interested in music.
The history of the area of triangles is unknown. The formulas of the area of triangles take place in Euclid archives. However, area knowledge has been known in Egypt before Euclid.
Dilek is looking to the ship on the top of the tower. The distance between Dilek and the ship is 25 meters. And the height of the tower is 7 meters. Şeyma is waiting to the ship on the floor of the tower. How many meters are there between Şeyma and the ship?
The famous Sierpinski Gasket constitutes an early awareness of fractal geometry. Sierpinski has been described as "the greatest and most productive of Polish mathematicians."
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THANK YOU FOR LISTENING
Dilek Koç
Şeyma Erbaş
Centroid: The point where the three medians of the triangle intersect.
The Incircle of a triangle: Also known as "inscribed circle", it is the largest circle that will fit inside the triangle. Each of the triangle's three sides is a tangent to the circle. It's the point where the angle bisectors meet.
The Circumcenter of a triangle: The point where the three perpendicular bisectors of a triangle meet.
Orthocenter of a Triangle: The point where the three altitudes of a triangle intersect.
References
http://illuminations.nctm.org/

http://www.cut-the-knot.org/pythagoras/index.shtml

http://www.math-aids.com/

http://en.wikipedia.org/wiki
http://illuminations.nctm.org/Activity.aspx?id=3567
If we bend the altitude from the middle and make a rectangle with bending base, we can find area easily. The area of rectangle is half area of the triangle. It is valid for every kind of triangles, because we are just bending middle sides. For each triangle we can get a rectangle as a half area of the triangle.
If we take two equal triangles and put them next to each other, we find a parallelogram. So the area of the triangle is half area of the parallelogram.
The sum of the areas of squares with sides b and a are equal to the area of square with side c. So;
Activity 2;
Activity 1;
Activity 1;
Activity 2;
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