SLOPE:
A number used for describing the line steepness and line direction in mathematics is known as a slope.
DENOTATION:
Mostly the slopes is denoted by the “m “letter.
CALCULATION OF SLOPE:
The slopes are calculated by determining the ratio of two distinct points at the same time. The ratio is between vertical differences to the horizontal difference.
FORMULA:
In mathematics, to find slopes m of any line use the formula given below:
m= y2-y1/ x2-x1
DEGREE OF STEEPNESS:
The greater absolute value of slopes indicates the more steepness of the slope
POSITIVE SLOPE:
If a line goes up from left to right so it is increasing and the slopes are considered positive.
m> 0
NEGATIVE SLOPE:
If a line goes down from left to right so it is decreasing and the slopes are negative.
m<0
IMPORTANCE OF SLOPE CONCEPT:
The concept of slopes is also very important in a different field. We can easily and directly apply the concept of slope to grades or gradients in civil engineering and geography. By using the formula below we can apply the concept of slope in finding the answers of various fields:
m= tan(θ)
RISING SLOPE:
Thus, an angle of 45 degrees of rising indicates that the slope is positive (+1)
FALLING SLOPE:
On the other hand side, an angle of 45 degrees of the falling line indicates that the slope is negative (-1)
ROLE OF SLOPE IN WRITING EQUATION OF LINE SLOPE:
For the writing equation of the line, the slope is necessary. As the equation of line slope is written typically as:
y= m .x +b
Where
m= slope
b= y- interception.
EXAMPLE:
For instance, consider a line running through the points (2,8) and (3,20) and this line has a slope, m,
20-8/3-2 =12
You can write it in slope form as written below:
y- 8=12(x-2) = 12x – 24
y= 12x-24-8
y= 12x- 16
Angle make between 90 and -90 degrees
θ = arctan (12)= 85.2 degrees
Consider 2 lines as y = −3x + 1 and y = −3x − 2.
According to them slope m= -3
That proves that they are parallel
If we consider these two lines: y = −3x + 1 and y = x/3 – 2 then
m1 = -3
While
m2 = 1/3
The product of this slope is -1 so they are perpendicular.
PYTHAGOREAN THEOREM:
It stated that the sum of the square of the other two sides of the triangle is equal to the square of the hypotenuse side of the right triangle.
FORMULA:
Pythagorean Theorem formula is written below:
a^2+b^2=c^2
Where
c= hypotenuse a and b = non-hypotenuse sides of the triangle.
HYPOTENUSE:
The hypotenuse is the longest side of the right triangle and it is the opposite side to the right angle. The other two sides in the right triangles meet at 90 degrees.
IMPORTANCE:
- Generally, the Pythagorean Theorem is used to find out the hypotenuse of the right triangle.
- Only for the right triangle, the Pythagoras theorem works so to determine whether the triangle is right or not. You can test by applying the Pythagoras theorem
- Pythagoras theorem helps in determining or finding the 3rd side of the triangle.
- Along with that Pythagoras theorem also helps in finding the length of missing sides of rectangular and square when triangles are pushed together.
- To build rectangular and square, the Pythagoras theorem plays a major part.
LENGTH OF SLOPE:
The length of the slopes can be calculated with the help of the Pythagoras theorem. The formula for finding slopes by using Pythagoras theorem is described below:
rise^ 2 + run ^2 = slope length^2
Where,
rise = vertical distance
run = horizontal distance
But instead of doing calculations, we may also use a slope finder tool that converts the manual calculation to auto by the tool.
EXAMPLE:
For example, a right triangle has three sides and you have to determine the hypotenuse which is the longest side.
A= 10
B= 12
C = unknown
SOLUTION:
By using Pythagorean Start with: a2 + b2 = c2
Put in what we know: 102 + 122 = c2
Calculate squares: 100+ 144 = c2
100+144=244:244 = c2
Swap sides:c2 = 244
The square root of both sides: c = √244
Calculate: c = 15.6
EXAMPLE 2:
You can also use the Pythagorean theorem to determine whether the triangle is right-angled or not for this purpose you need to follow the procedure illustrated below;
Does a 4, 15, 16 triangles have a Right Angle?
Does 42 + 152 = 162 ?
- 42 + 152 = 24 + 225 = 249,
- but 162 = 256
So this indicates that it is not a right triangle.
If you want to find out hypotenuse use the formula given below:
c = √(a2 + b2)
This formula will help you in finding the hypotenuse. But for the slope of calculation of the sides, we can try the technology-based online tools like Pythagorean theorem solver. This will help us to find any side of the triangle and also we can recheck whether we find the right answer or not.
So this is all about the slope and the Pythagorean theorem which are the most useful concepts for students in their learning. So keep yourself attached to learning with DewArticle.
