STD X – GEOMETRY – NEWTON

SIMILARITY

“A line drawn parallel to any side of a triangle, divides the other two sides in the same ratio.”

Have you thought what similarity with respect to two triangles is? Well, Two triangles are said to be similar if their corresponding angles are equal and corresponding sides are proportional.  By using AAA similarity theorem, SSS similarity theorem and SAS similarity theorem we can prove two triangles are similar.

We now know that two triangles are similar if their corresponding angles are equal, and the corresponding sides are proportional. If two triangles are similar, then the ratio of their areas is equal to the ratio of the squares of their corresponding sides.We shall learn more about theorems related to similar triangles in this topic.

LOCI

The concept of locus is very important in geometry. Suppose X and Y are two fixed point in two dimensional co-ordinate plane. If a point M moves on this plane in such a manner that its distance from the points X and Y are always equal, then the point M will trace out a definite path on the plane. Thus, a moving point M trace out a definite path on the given plane if it satisfies some specified geometrical conditions. Such a path trace out by a moving point M on a plane is called its locus. We shall learn more about the locus and theorems based on it.

CIRCLES – CHORD PROPERTIES

A chord of a circle is a straight line segment whose endpoints both lie on the circle. A line that joins two points on the circumference of a circle is called a chord. A chord that passes through a circle’s centre point is the circle’s diameter. Every diameter is a chord, but not every chord is a diameter.

CIRCLES – TANGENT AND INTERSECTING CHORDS

The word tangent comes from the Latin word ‘tangere’, which means to touch. A tangent is a line which touches or intersects the circle at only one point. The point where the tangent meets the circle is called the point of contact or the point of tangency.

• A tangent line to a circle is a line that touches the circle at exactly one point. 
• A tangent to a circle is perpendicular to the radius at the point of tangency.

CONSTRUCTION

Construction” in Geometry means to draw shapes, angles or lines accurately. These constructions use only compass, straightedge (i.e. ruler) and a pencil.

Let us learn some constructions involving circles in the current topic

• Constructing a tangent to a circle from an external point
• Constructing circumcircle of a triangle
• Constructing incircle of a triangle
• Construct a circle circumscribing a given regular hexagon