GEOMETRY

TRIANGLE

•         A triangle is a plane shape bounded with three sides.

•         A Triangle has three edges, three vertices, and three (3) interior angles.

•         A triangle also have three altitudes (that is, heights).

•         Triangles can be classified based on the nature of its angles, or based on the nature of its sides.

•         Classification based on the nature of Angles: Acute-angled Triangle ( all three angles are acute angles), Right-angled Triangle (one angle is a right-angle), and Obtuse-angled Triangle (one angle is an obtuse angle).

•         Classification based on the nature of side lengths: Equilateral Triangle (all three sides are equal in length), Isosceles Triangle (two side lengths are equal) and Scalene triangle (no two side lengths are equal).

Given a triangle with side lengths

•         If the triangle is an Acute-angled Triangle.

•         If the triangle is a Right-angled Triangle.

•         If the triangle is an Obtuse-angled Triangle.

•         Note that the above relationships are between the squares of the side lengths.

•         The Triangle Inequality simply states that in any triangle that truly exist, that is, a non-degenerate triangle (a non-degenerate triangle is a triangle with positive area), the sum of the side lengths of any two sides must be greater that the length of the third side.

•         Given a triangle with side lengths and the triangle inequality says (that is, the sum of the two shorter sides must be greater than the longest side).

•         The Triangle inequality also states that in any polygon, it’s semi-perimeter is greater than the length of any side.

•         Given a triangle with side lengths a, b, c; by triangle inequality we know that the sum of any two side lengths is larger than the third side

Adding to both sides

•         Given an n-sided polygon with sides . We know from triangle inequality that any side length is less than the sum of all other side lengths.

Adding to both sides

•         Thus, from triangle inequality we know that the semi-perimeter of any polygon is greater than any side length of that polygon.

There are many ways to find the area of a triangle. Some include:

•         When we are given all three sides of the triangle.

•         When we are given two sides and an included angle.

•         When we are given a side length and it’s altitude.

•         When we are given the semi-perimeter and in-radius of the triangle.

•         When we are given all three side lengths and the circum-radius of the triangle.

•         Given Triangle ABC with side lengths

•         Area of triangle

•         Area of triangle =

•         Area of triangle

•         Every side of a triangle has an altitude (also known as height).

•         An altitude is a line drawn from a vertex, perpendicular to the opposite side.

•         Area of a triangle equals half the product of a side multiplied to it’s altitude.

Problem 1

In a triangle with integer side lengths, one side is three times as long as a second side, and the length of the third is 15. what is the greatest possible perimeter of the triangle? (AMC 10, 2006)

Solution

Let be the length of the first side; this results in the length of the second side as

The lengths of the triangle are

The longest side of this triangle is either

When is the longest side, we know from triangle inequality that

The greatest value of here is 7.

When 15 is the longest side, we know from triangle inequality that

The greatest value of here is 4, this is because

Thus, the greatest possible perimeter of the triangle is gotten when

Perimeter

Greatest possible perimeter

Problem 2

Find the area of a triangle with lengths 7cm, 24cm and 25cm.

Solution

Area

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