Why Heron found another formula for the area of triangle while a formula ( 1/2 *base *altitude )is already existing ?

Heron's formula is named after Hero of Alexendria, a Greek Engineer and Mathematician in 10 - 70 AD. You can use this formula to find the area of a triangle using the 3 side lengths. Therefore, you do not have to rely on the formula for area that uses base and height. Heron's formula is a formula that helps you to find the area of a triangle when all you know are the lengths of the sides. It is an ancient formula that has been around for a couple thousand years!

If X=-2,Y=6 is solution of equation 3ax+2by=6 then find the value of b from 2(a-1)+2(3b-4)=4

Substituting x = 2 and y = 6 in eqn,we get
3 * a * 2 + 2 * b * 6 = 6
6a + 12b = 6
ie a + 2b = 1
so, a = 1 - 2b
Substituting a = 1 - 2b in second equation,we get
2(1 - 2b - 1) + 2(3b - 4) = 4
-4b + 6b - 8 = 4
2b = 12
b = 6

ABCD is a square and ABE is an equilateral triangle outside the square. Prove that angle ACE =half of angle ABE.

As ABE is an equilateral triangle, hence

$∠ABE = 60^{0}$ .......(i)

$AB =BE$

And ABCD is a square hence

$∠ABC = 90^{0}$

$AB =BC $........(ii)

By (i) and (ii) we get

$BC = BE$

In triangle $BCE$

$∠BEC = ∠BCE$ [angles opposite to equal sides are equal]

Also

$ \Longrightarrow ∠BEC + ∠BCE + ∠CBE = 180^{0}$ [Angle sum property]

$\Longrightarrow ∠BCE + ∠BCE + (∠ABC + ∠ABE) = 180^{0}$

$2\Longrightarrow ∠BCE + (90^{0} + 60^{0}) = 180^{0}$

$2\Longrightarrow ∠BCE = 180^{0} - 150 ^{0}$

$\Longrightarrow ∠ BCE = 15^{0}$

As diagonals of the ...

Difference between a Cube and cuboid.

Cuboid
A cuboid is a box-shaped object. It has six flat faces and all angles are right angles. And all of its faces are rectangles.
Cube
When all three lengths are equal it is called a cube (or hexahedron) and each face is a square.

Is 0 rational

Rational numbers are the numbers which can be expressed in form of p/q where p & q both are integers and q not equal to 0. We know 0 can be written as 0/1, and observe that both 0 & 1 are integers and the denominator i.e. ‘1’ ? 0. So we conclude that ‘0’ is a rational number and not irrational. All whole numbers, all Positive and Negative Numbers are rational numbers.

What is the number of lines segment possible with three collinear point?

" Three or more points lie on the same line, then the points are called Collinear Points" . It is possible to draw one and only one line segment passing through the given three given points.

A rectangular park 60 m long and 40 m wide has two concrete crossroads running in the middle of the park and rest of the park has been used as a lawn. If the area of the lawn is 2109 sq. m, then what is the width of the road?

Area of the park = (60 x 40) m² = 2400 m².
Area of the lawn = 2109 m².
Area of the crossroads = (2400 - 2109) m² = 291 m².
Let the width of the road be x metres. Then,
60x + 40x - x² = 291
x² - 100x + 291 = 0
(x - 97)(x - 3) = 0
x = 3.

The average of 20 numbers is zero. Of them, at the most, how many may be greater than zero?

Average of 20 numbers = 0.
Sum of 20 numbers (0 x 20) = 0.
It is quite possible that 19 of these numbers may be positive and if their sum is a then 20th number is (-a).

How a line is different from a line segment.

Line -segment : A part or portion of a line with two end points is called a line-segment. A line-segment $AB$ is denoted by $ \overline{AB} $ where, $A$ and $B$ are the two end points.
Line : A line-segment extended indefinitely in both the directions is called a line. A line is denoted by $ \overleftrightarrow{AB} $. A line has no end point.

Find the smallest 4-digit number which in divisible by 18,24 and 32.

The smallest 4-digit number which is divisible by 18, 24 and 32., will be simply the 4 digit number necessarily a multiple of LCM of the given numbers i.e. 18, 24 and 32. LCM of 18, 24 and 32= 2 ×2 ×2 ×2 ×2 ×3×3× = 288 ? required smallest 4-digit number which is divisible by 18, 24 and 32 number is = 4 × 288 = 1152.