GEOMETRY
Category : 8th Class
Introduction:
The word ‘geometry originally came from the Greek word ‘geo’ meaning ‘earth’ and ‘metron’ meaning ‘measurement’. Therefore the word geometry means ‘measurement of earth’ or is the science of properties and relations of figures.
The scope of plane geometry, as a branch of mathematics, has broadened the study about plane figures-line, angles, triangles, quadrilaterals, circle, etc.
POINTS TO REMEMBER
s and t are two transversal intersecting three parallel lines 1, m and n at A, B, C and P, Q, R, respectively
\[ABCD=\frac{1}{2}(AB+D\times CE)\] \[\angle R={{138}^{o}}\]
PARALLEL LINE AND TRANSVERSAL
If a transversal P intersect two parallel lines e|| m (as shown in the figure) then,
(i) Corresponding angles are equal
\[\angle R={{138}^{o}}\] {corresponding angles}
\[\angle ACB={{65}^{o}}\] {corresponding angles}
(ii) Alternate interior angles are equal
\[\angle ABC\] {alternate interior angles}
\[{{25}^{o}}\] {alternate interior angles}
(iii) Sum of consecutive interior angles in the same side of transversal is \[{{35}^{o}}\]
\[{{55}^{o}}\]
\[{{208}^{o}}\]
In the figure given below \[{{52}^{o}}\]
TRLANGLE
A triangle is a closed plane figure bound by three line segments. So, A triangle has six parts, three sides AB/BC and CA three angles \[\Delta XYZ\] and \[\Delta XYZ\]
Sum of all the angles in a triangle is \[{{60}^{o}}\]
\[{{30}^{o}}\] \[{{80}^{o}}\]
Median: The line segment joining a vertex of a triangle to the mid-point of the opposite side is called a median of the triangle.
(i) Intersecting point of the medians is called the centroid o the triangle.
(ii) Centroid divide the median in the ratio 2:1
AD is the median \[{{100}^{o}}\]\[\angle DAC={{54}^{o}}\](median bisect opposite side)
AD, BE and CF are medians\[\angle ACB={{63}^{o}}\]O is centroid
So, \[{{(2.5)}^{2}}+{{(PT)}^{2}}={{(4.5)}^{2}}\] or \[\angle ACB={{63}^{o}}\]
QUADRILATERAL
A closed figure with four sides is called quadrilateral
Quadrilateral ABCD has:
(i) Four sides: AB,BC, CD and DA.
(ii) Four vertices: A, B,C and D.
(iii) Four angles: \[{{(PT)}^{2}}=20.25-6.25=14\] and \[{{(PT)}^{2}}=PA\times PB\].
(iv) Two diagonals: AC and BD.
Properties of Quadrilateral
TYPES OF QUADRILATERAL
TRAPEZUM
A quadrilateral in which one pair of opposite sides are parallel is called trapezium.
In quadrilateral (ABCD), \[PA=(x+3)\]
Thus, ABCD is a trapezium.
PARALLELOGRAM
A quadrilateral with both pair of opposite sides are parallel is called a parallelogram.
Properties of Parallelogram
(i) \[14=(x+3)x\]
(ii) \[\Rightarrow \]
(iii) \[{{x}^{2}}+3x-14=0\]
(iv) \[\Rightarrow \]
RHOMBUS
A parallelogram with a pair of adjacent sides are equal is called a rhombus.
Properties of Rhombus: All the sides of a rhombus are equal
OA = OC
OB = OD
Diagonals of a rhombus bisects each other at \[PB=x=\frac{-3\pm \sqrt{65}}{2}=\frac{-3+8.06}{2}=2.53\]
Properties of Rectangle
\[\therefore \]
\[\angle CYA=\angle CYA={{40}^{o}}\] and \[\angle AXB={{30}^{o}}\]
\[AB=XB\] and \[\therefore \]
Diagonals AC and BD bisect each other.
SQUARE
A rectangle with a pair of adjacent sides equal is a square.
Properties of Square
CIRCLES
A circle is a simple closed curve, all the points of which are at the same distance from a given fixed point.
PARTS OF ACIRCLE
Centre: The fixed point in the plane which is equidistant from every point on the boundary of the circle is called centre. In the adjoining figure, 0 is the centre of the circle.
Radius: The fixed distance between the centre and any point of the circle is called radius. In figure, \[\angle AXB=\angle XAB={{30}^{o}}\] is a radius.
Chord: A line segment joining any two points on a circle is called a chord of the circle. In figure. \[\angle XAY={{180}^{o}}-({{40}^{o}}+{{30}^{o}})={{110}^{o}}\] is a chord.
Diameter: A chord that passes through the centre of a circle is called diameter of the circle. In figure, AB is a diameter. The length of a diameter \[\because \]radius. In a circle, diameter is the longest chord.
Circumference: The distance around a circle is called the circumference. Circumference of a circle is the perimeter of that circle.
Arc: Apart of a circumference is called an arc. In the above figure, the curve line AR is an arc of the circle. It is written as AR.
Properties of Circle
.
If \[\angle BAC={{40}^{o}}.\] \[AN\bot BC\] \[DE:BC=3:5\]
If \[\frac{AM}{AN}=\frac{3}{5}(Also)\]\[\Delta ADE=\frac{1}{2}(DE)(AM)=\frac{1}{2}(3)(3)=9/2\]\[\Delta ABC=\frac{1}{2}(BC)(AN)=\frac{1}{2}(5)(5)=\frac{25}{2}\]
If \[\Delta ABC-Area\]\[\Delta ADE=\frac{25}{2}-\frac{9}{2}=\frac{16}{2}\] \[\therefore \]
If \[{{40}^{o}}\]\[{{60}^{o}}\] \[{{90}^{o}}\]
\[{{30}^{o}}\]
Angle subtended by the arc AB at the centre is \[{{45}^{o}}\] and the angle subtended by the same arc to the remaining part of the circle is \[{{60}^{o}}\] .
·
\[\angle CAD={{50}^{o}}\]is a cyclic quadrilateral
\[\angle BED={{120}^{o}}\] \[\angle BCD\] and \[{{75}^{o}}\]
\[{{85}^{o}}\]\[{{60}^{o}}\] is angle of semicircle.
\[\angle BAC={{45}^{o}}\] \[\angle BED={{120}^{o}}\]\[\angle ABD\]
\[\angle R={{138}^{o}}\] \[\angle ACB={{65}^{o}}\]
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