¶ Geometry
Triangles:
Ratios of sides:
If the triangle is a right-angled triangle with angles measuring 30:60: 90, the opposite sides will also maintain a ratio of 1:√3: 2.
If the triangle is an isosceles right-angled triangle measuring 45:45:90, the opposite sides will
maintain a ratio of 1:1:√2.
The speciality of 30-60-90 Triangles:
Finding ratio starting with odd integers in Pythagorean triplets:
Square the odd integer (first one) and divide it by 2. The integers precedent and subsequent to it are the two required numbers of the triplets. For example, 3, 4, 5 is a Pythagorean triplet. If we select the value 3 and square it, we get 32=9. Then we need to divide it by 2. So, 9/2= 4.5. We know that 4<4.5<5. So, if 3 is one of the values of a Pythagorean triplet, the other two will be 4 & 5.
Distribution of Areas:
When the medians of 2 sides intersect and the midpoints of those two meet to form a triangle the area is distributed as following:
The ratio of newly formed triangles inside will follow the ratio of 1:2:2:3:4
Some Basics on Geometry:
Polygons:
One angle of a regular polygon= (n - 2) *180/n
Here, n= no. of sides in a polygon
Interior & Exterior Angles:
Circles:
Area of Arc = (Θ/360)*πr2
Length of the arc = (Θ/360)× 2πr.
Here, Θ=angle subtended by the arc from the Centre.
Rotation of Wheel:
r1n1= r2n2 [finding rotations/ radius of one wheel based on another]
Here,
r= radius of the wheels
n= number of rotations
Coordinate Geometry:
Midpoints of a line:
x= (x1+x2)/2 y= (y1+y2)/2
Length of a line:
√ {(x1-x2)2+ (y1-y2)2}
Slope = (y2-y1)/(x2-x1)
Clock related problems:
For finding the angles between the hands of a clock:
|(11/2)× minute hand - 30× hour hand |
Some theorems:
Angles:
01. The sum of two adjacent angles which makes a straight line is equal to two right angles or 180°.
02. When two straight lines intersect, the vertically opposite angles are equal.
03. When two parallel straight lines are intersected by a straight line:
a. The pair of exterior and interior alternate angles are equal
b. The pair of interior angles on the same side of the transversal are supplementary.
c. Pairs of corresponding angles are equal.
04. The lines which are parallel to a given line are parallel to each other.
Angles and Arcs:
Triangles:
01. The sum of the three angles of a triangle is equal to 180°.
02. If one side of a triangle is extended and an exterior angle is formed then the exterior angle will be equal to the sum of two opposite interior angles.
03. The acute angles of a right angle triangle are complementary to each other.
04. If two sides of a triangle are equal, the angles opposite to the equal sides are also equal (vice versa).
05. If one side of a triangle is greater than the other, the side opposite the greater angle is greater than that of the lesser.
06. The sum of the lengths of any two sides of a triangle is greater than the third side.
07. The line segment joining the mid-points of any two sides of a triangle is parallel to the third and half in length.
08. If two triangles are similar, their corresponding sides are proportional. (vice versa)
09. If one angle of one triangle is equal to an angle of another triangle and the side adjoining the equal angles are proportional, the two triangles will be similar.
10. If the two angles of one triangle are equal to the two angles of the other, the two triangles are similar.
11. The ratio of the areas of the two similar triangles is equal to the ratio of the areas of the squares drawn of their corresponding sides.
Let's Know About Triangles:
More on Triangles:
Circles:
01. The perpendicular drawn from the Centre of the circle to a chord, bisects the chord (vice versa).
02. A straight line cannot intersect a circle in more than two points.
03. All equal chords of a circle are equidistant from the Centre. (vice versa)
04. The diameter is the greatest chord of the circle.
05. The angle at the Centre is double the angle at the circumference.
06. The angle in the semi-circle is a right triangle.
07. The circle drawn with hypotenuse of a right-angled triangle as diameter passes through the vertices of a triangle.
08. At any point of the circle, the perpendicular to the radius is a tangent to the circle.
Circles:
Inscribed Circles and Squares: