## VERTICAL AND SPECIAL TRIANGLES

In this section we will see the relations between the sides of right triangles. We will also examine the most used shapes of right triangles as special triangles..

We will also consider isosceles and equilateral triangles using special triangles in this section.

Since this information will be used in all areas of geometry, it is useful to learn it well.

A triangle whose angle is 90 ° is called a right triangle.

In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the perpendicular sides. This relation is called the Pythagorean Relation.

In a right triangle, the median length of the hypotenuse is equal to half the length of the hypotenuse.

In a right triangle, if the dimension of one of the acute angles is 30 °, the perpendicular side length opposite this angle is equal to half the length of the hypotenuse.

In a right triangle, the height of the hypotenuse divides the triangle into each other and into two triangles similar to the triangle itself.

Euclidean Theories

Height relation: In a right triangle, the length of the height of the hypotenuse is the geometric middle between the lengths of the line segments that the height separates on the hypotenuse. (H² = p.k)

Right side relation: In a right triangle, each right side is the geometrical middle of the part separated by the hypotenuse and the height from the hypotenuse on its side. (C ² = p.a), (b² = k.a)

TWIN SIDES TRIANGLE

It is called a triangle whose two sides are equal. ABC (| AB | = | AC |) in isosceles triangle, m (B) = m (C). the point is called the peak point.

In an isosceles triangle, the bisector of the base is the same line segment as the bisector of the height and vertex angle.

In an isosceles triangle, the line segment joining the midpoint of the base to the vertex angle is perpendicular to the base.

In an isosceles triangle, the sum of the posts lowered from one point to the equal sides from the base is equal to the height of the equal sides.

In an isosceles triangle, the sum of the lengths of the parallel line segments drawn from one point to the equilateral sides from the base is equal to the length of one of the equilateral sides.

EQUILATERAL TRIANGLE

A triangle with equal dimensions of all sides, internal angles and exterior angles is called a triangle.

The sum of the lengths of the perpendiculars lowered to the sides of the triangle from a point taken through an equilateral triangle is equal to the height of the equilateral triangle.

The sum of the lengths of the parallel line segments drawn from any point in an equilateral triangle to the sides is equal to the length of one side of the triangle.

The center of gravity in an equilateral triangle; it is the center of both the inner tangent circle and the circumferential circle of the triangle.

* Vertical and Special Triangles *There are 5 tests each consisting 8 questions.

**WARNING:**

If there is any angle of 30, 45, 60 in a triangle, a special triangle is created by drawing perpendicular to these angles and the question is solved.

Also; If there are any of the angles 120, 135, 150 in a triangle, a special triangle is created by drawing the outer angles of these angles 60, 45, 30 perpendicularly.