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Triangles

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A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ABC.

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Types of triangles

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**By relative lengths of sides**

Triangles can be classified according to the relative lengths of their sides: In an equilateral triangle, all sides have the same length. An equilateral triangle is also a regular polygon with all angles measuring 60°.

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**In an isosceles triangle, two sides are equal in length**

In an isosceles triangle, two sides are equal in length. An isosceles triangle also has two angles of the same measure; namely, the angles opposite to the two sides of the same length.

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**In a scalene triangle, all sides and angles are different from one another.**

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By internal angles A right triangle (or right-angled triangle, formerly called a rectangled triangle) has one of its interior angles measuring 90° (a right angle). The side opposite to the right angle is the hypotenuse; it is the longest side in the right triangle. The other two sides are the legs or catheti (singular: cathetus) of the triangle. Right triangles obey the Pythagorean theorem: the sum of the squares of the lengths of the two legs is equal to the square of the length of the hypotenuse: a2 + b2 = c2, where a and b are the lengths of the legs and c is the length of the hypotenuse. Special right triangles are right triangles with additional properties that make calculations involving them easier.

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A triangle that has all interior angles measuring less than 90° is an acute triangle or acute-angled triangle.

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A triangle that has one angle that measures more than 90° is an obtuse triangle or obtuse-angled triangle.

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Area of triangles

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**Area of a triangle The area is half of the base times height.**

"b" is the distance along the base "h" is the height (measured at right angles to the base) Area = ½bh

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Another way of writing the formula is bh/2 Example: What is the area of this triangle? Height = h = 12 Base = b = 20 Area = bh/2 = 20 × 12 / 2 = 120

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**Why is the Area "Half of bh"?**

Imagine you "doubled" the triangle (flip it around one of the upper edges) to make a square-like shape (it would be a "parallelogram" actually), Then the whole area would be bh (that would be for both triangles, so just one is ½bh), like this: You can also see that if you sliced the new triangle and placed the sliced part on the other side you get a simple rectangle, whose area is bh.

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The End

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Copyright © Cengage Learning. All rights reserved. 12 Geometry.

Copyright © Cengage Learning. All rights reserved. 12 Geometry.

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