Introduction: Triangles and circles

What is a triangle?

A triangle can be defined as a polygon which has three angles and three sides. The interior angles of a triangle sum up to 180 degrees and the exterior angles sum up to 360 degrees. Depending upon the angle and its length, a triangle can be categorized in the following types-

1.Scalene Triangle –
All the three sides of the triangle are of different measure.

2.Isosceles Triangle –
Any two sides of the triangle are of equal length.

3.Equilateral Triangle –
All the three sides of a triangle are equal and each angle measures 60 degrees.

4.Acute angled Triangle –
All the angles are smaller than 90 degrees.

5.Right angle Triangle –
Anyone of the three angles is equal to 90 degrees.

6.Obtuse-angled Triangle –
One of the angles is greater than 90 degrees.

PARTS OF TRIANGLE

RIGHT ANGLE TRAINGLE

Similarity criteria of polygons having same number of sides-

1.Their corresponding angles are equal, and
2.Their corresponding sides are in the same ratio (or proportion)

Similarity criteria of triangles –

Side-Side- Side (SSS) Similarity Criterion – 

When the corresponding sides of any two triangles are in the same ratio, then their corresponding angles will be equal and the triangle will be considered as similar triangles.

Angle Angle Angle (AAA) Similarity Criterion –

When the corresponding angles of any two triangles are equal, then their corresponding side will be in the same ratio and the triangles are considered to be similar.

Angle-Angle (AA) Similarity Criterion – 

When two angles of one triangle are respectively equal to the two angles of the other triangle, then the two triangles are considered as similar.

Side-Angle-Side (SAS) Similarity Criterion –

When one angle of a triangle is equal to one angle of another triangle and the sides including these angles are in the same ratio (proportional), then the triangles are said to be similar.

 Pythagoras theorem –

 “In a right-angled triangle, the sum of squares of two sides of a right triangle is equal to the square of the hypotenuse of the triangle.”

AC² = AB² + BC²

Circle and the line in the plane-

• they can be non-intersecting
• they can have a single common point: in this case, the line touches the circle.
• they can have two common points: in this case, the line cuts the circle.

Tangent-

A tangent to a circle is a line that touches the circle at exactly one point. For every point on the circle, there is a unique tangent passing through it.

Secant-

A circle’s secant is a line that shares two points with the circle. It divides the circle in two parts, forming a circle chord.

Tangent perpendicular to the radius at the point of contact-

The theorem states that “the tangent to the circle at any point is the perpendicular to the radius of the circle that passes through the point of contact”

The number of tangents drawn from a given point-

If the point is in the circle’s interior, any line that passes through it is a secant. As a result, no tangent can be traced to a circle that goes through a point within it.

There is exactly one tangent to a circle that passes through a point of intersection on the circle.

There are exactly two tangents to a circle through a point that is outside of the circle.

Length of a tangent-

The length of the tangent from the point (Say P) to the circle is defined as the segment of the tangent from the external point P to the point of tangency I with the circle. In this case, PI is the tangent length.

Lengths of a tangents drawn from a external points-

Two tangents are of equal length when the tangent is drawn from an external point to a circle.