Geometry Tutorial Part - 5 - Campus Placements

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Geometry Tutorial Part - 5

Author: Sureshbala

RHOMBUS

A rhombus is a parallelogram in which adjacent sides are equal(all four sides of a rhombus are equal)

Since rhombus is a parallelogram, all the properties of a parallelogram apply to a rhombus. Further, in a rhombus, the diagonals bisect each other perpendicularly.

Conversely, any quadrilateral where the two diagonals bisect each other at right angles will be a rhombus.

The four triangles that are formed by the two bisecting diagonals with the four sides of the rhombus will all be congruent. In Fig. 1.28, the four triangles PAB, PBC, PCD and PAD are congruent.

$Side\; of\; a\; rhombus = \sqrt {1/4 *Sum \, of \, squares \, of \,the \,diagonals}$

RECTANGLE

A rectangle also is a special type of parallelogram and hence all properties of a parallelogram apply to rectangles also. A rectangle is a parallelogram in which two adjacent angles are equal or each of the angles is equal to $90^{\circ}$

The diagonals of a rectangle are equal (and, of course, bisect each other.)

When a rectangle is inscribed in a circle, the diagonals become the diameters of the diameters of the circle.

If a and b are the two adjacent sides of a rectangle, then the diagonal is given by $\sqrt {a^2+b^2}$

If a rectangle and a triangle are on the same base and between the same parallels, then the area of the triangle will be equal to half the area of the rectangle.

SQUARE

A square is a rectangle in which all four sides are equal (or a rhombus in which all four angles are equal, i.e., all are right angles) Hence, the diagonals are equal and they bisect at right angles. So, all the properties of a rectangle as well as those of a rhombus hold good for a square.

$Diagonal = \sqrt{2}*Side$

When a circle is inscribed in a circle, the diagonals become the diameters of the circle.

The largest triangle that can be inscribed in a given circle will be a square.

POLYGON

Any closed figure with three or more sides is called a polygon.

A convex polygon is one in which each of the interior angles is less than . It can be noticed that any straight line drawn cutting a convex polygon passes only two sides of the polygon, as shown in the figure below.

In a concave polygon, it is possible to draw lines passing through more than two sides, as shown in the figure below.

A regular polygon is convex polygon in which all sides are equal and all angles are equal. A regular polygon can be inscribed in a circle. The centre of the circumscribing circle (the circle in which the polygon is inscribed) of a regular polygon is called the centre of the polygon.

The names of polygons with three, four, five, six, seven, eight, nine and ten sides are respectively triangle, quadrilateral, pentagon, hexagon, heptagon, octagon, nonagon and decagon.

The sum of interior angles of a convex polygon is equal to (2n-4) right angles where n is the number of the sides of the polygon

If each of the sides of a convex polygon is extended the sum of the external angles thus formed is equal to 4 Right Angles (i.e $360^{\circ}$ )

In a regular polygon of n sides, if each of the interior angles is , $d^{\circ}$

Then $d= \cfrac {2n-4}{n}*90^{\circ}$ and each exterior angle $=\cfrac {360^{\circ}}{n}$

It will be helpful to remember the interior angles of the following regular polygons:

Regualr Pentagon : $108^{\circ}$

Regualr hexagon : $120^{\circ}$

Regualr octagon : $135^{\circ}$

If the centre a regular polygon (with n sides) is joined with each of the vertices, we get n identical triangles inside the polygon. In general, all these triangles are isosceles triangles. Only in case of a regular hexagon, all these triangles are equilateral triangles, i.e., in a regular hexagon, the radius of the circumscribing circle is equal to the side of the hexagon.

A line joining any two non-adjacent vertices of a polygon is called a diagonal. A polygon with n sides will have $\cfrac {n(n-3)}{2}$ diagonals

CIRCLE

A circle is a closed curve drawn such that any point on the curve is equidistant from a fixed point. The fixed point is called the centre of the circle and the distance from the centre to any point on the circle is called the radius of the circle.

Diameter is a straight line passing through the centre of the circle and joining two points on the circle. A circle is symmetric about any diameter.

A chord is a point joining two points on the circumference of a circle (AB inFig.1.30). Diameter is the largest chord in a circle

A secant is a line intersecting a circle in two distinct points and extending outside the circle also.

A line that touches the circle at only one point is a tangent to the circle ( is a tangent touching the circle at the point R in Fig. 1.30).

If PAB and PCD are two secants (in Fig. 1.29), then PA.PB = PC.PD

If PAB and PCD are secants and PT is a tangent to the circle at T (in Fig. 1.29), Then PA.PB = PC.PD=PT^2

Two tangents can be drawn to the circle from any point outside the circle and these two tangents are equal in length. In Fig. 1.30, P is the external point and the two tangents PX and PY are equal.

A perpendicular drawn from the centre of the circle to a chord bisects the chord (in Fig. 1.30, OC, the perpendicular from O to the chord AB bisects AB) and conversely, the perpendicular bisector of a chord passes through the centre of the circle.

Two chords that are equal in length will be equidistant from the centre, and conversely two chords which are equidistant from the centre of the circle will be of the same length.

One and only one circle passes through any where given non-collinear points

When there are two intersecting circles, the line joining the centers of the two circles will perpendicularly bisect the line joining the points of intersection. In Fig. 1.31, the two circles with centers X and Y respectively intersect at the two points P and Q. The line XY (the line joining the centers) bisects PQ (the line joining the two points of intersection).

Two circles are said to touch each other if a common tangent can be drawn touching both the circles at the same point. This is called the point of contact of the two circles. The two circles may touch each other internally (as in Fig. 1.32) or externally (As in Fig. 1.33).when two circles touch each other, then the point of contact and the centers of the two circles are collinear, i.e., the point of contact lies on the line joining the centers of the two circles.

If two circles touch internally, the distance between the centers of the two circles is equal to the difference in the radii of the two circles. When two circles touch each other externally, then the distance between the centers of the two circles is equal to the sum of the radii of the two circles.

A tangent drawn to two circles is called a common tangent. In general, for two circles, there can be anywhere from zero to four common tangents drawn depending on the position of one circle in relation to the other.

If the common tangent is either parallel to the line of centers or cuts the line joining the centers not between the two circles but on one side of the circles, such a common tangent is called a direct common tangent. A common tangent that cuts the line joining the centers in between the two circles is called transverse common tangent.

If two circles are such that one lies completely inside the other (without touching each other), then there will not be any common tangent to these circles (refer to Fig. 1.34).

Two circles touching each other internally (i.e., still one circle lies inside the other), then there is only one common tangent possible and it is drawn at the point of contact of the two circles (refer to Fig. 1.32).

Two intersecting circles have two common tangents. Both these are direct common tangents and the two intersecting circles do not have a transverse common tangent (refer to Fig. 1.35).

Two circles touching each other externally have three common tangents. Out of these, two are direct common tangents and one is a transverse common tangent. The transverse common tangent is at the point of contact (refer to Fig. 1.33).

Two circles which are non-intersecting and non-enclosing (i.e. one does not lie inside the other) have four common tangents- two direct and two transverse common tangents (refer to Fig.1.36).

If $r_1\; and\; r_2$ are the radii of the two non-intersecting non-enclosing circles,

$Length\; of\; the\; direct\; common\; tangent =\sqrt{(Distance\; Between\; cetner)^2-(r_1-r_2)^2}$ $Length\; of\; transverse\; common\; tangent =\sqrt{(Distance\; Between\; cetner)^2-(r_1+r_2)^2}$

Two circles are said to be concentric if they have the same center. As is obvious, here the circle with smaller radius lies completely with in the circle with bigger radius.

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