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The vertices of a triangle are (2,1),(5,2)and(4,4).find the length of the perpendicular from the first vertex to the opposite side.?
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The vertices of a triangle are (2,1),(5,2)and(4,4).find the length of ...
A=(x1, y1) = (2,1)
B=(x2,y2)=(5,2)
C=(x3,y3)=(4,4)
First find length of sides of triangle
Formula :d=√((x2-x1)^2 + (y2-y1)^2)
A=(2,1)
B=(5,2)
AB=√((5-2)^2 + (2-1)^2)
so, AB=3.162
B=(5,2)
C=(4,4)
now,
BC=√((4-5)^2 + (4 - 2)^2)
so, BC=2.236
again similarly, AC=3.605
Now to find area of triangle
a = 3.162
b = 2.236
c =3.605
area=√(s(s-a)(s-b)(s-c))
s=( a+b+c)/2
s=(3.162+2.236+3.605)/2 = 4.5015
so, area= √(4.5015(4.5015-3.162)(4.5015-2.236)(4.5015-3.605))
=3.4995
The base corresponding to point A is BC
So, To find length of the perpendicular from the first vertex to the opposite side
Area of triangle =
1/2(base x height)
or, 1/2(2.236 x height) = 3.4995
therefore , height=3.13unit
Hence the length of the perpendicular from the first vertex to the opposite side is 3.13 units
B=(x2,y2)=(5,2)
C=(x3,y3)=(4,4)
First find length of sides of triangle
Formula :d=√((x2-x1)^2 + (y2-y1)^2)
A=(2,1)
B=(5,2)
AB=√((5-2)^2 + (2-1)^2)
so, AB=3.162
B=(5,2)
C=(4,4)
now,
BC=√((4-5)^2 + (4 - 2)^2)
so, BC=2.236
again similarly, AC=3.605
Now to find area of triangle
a = 3.162
b = 2.236
c =3.605
area=√(s(s-a)(s-b)(s-c))
s=( a+b+c)/2
s=(3.162+2.236+3.605)/2 = 4.5015
so, area= √(4.5015(4.5015-3.162)(4.5015-2.236)(4.5015-3.605))
=3.4995
The base corresponding to point A is BC
So, To find length of the perpendicular from the first vertex to the opposite side
Area of triangle =
1/2(base x height)
or, 1/2(2.236 x height) = 3.4995
therefore , height=3.13unit
Hence the length of the perpendicular from the first vertex to the opposite side is 3.13 units
Community Answer
The vertices of a triangle are (2,1),(5,2)and(4,4).find the length of ...
To find the length of the perpendicular from the first vertex to the opposite side of a triangle, we can follow these steps:
Step 1: Plot the given vertices on a coordinate plane.
The given vertices of the triangle are (2,1), (5,2), and (4,4). Let's plot these points on a coordinate plane.
Step 2: Find the equation of the line passing through the second and third vertices.
To find the equation of the line passing through (5,2) and (4,4), we can use the slope-intercept form of a line, which is y = mx + b.
First, let's find the slope (m):
m = (y2 - y1) / (x2 - x1)
= (4 - 2) / (4 - 5)
= 2 / -1
= -2
Now, let's find the y-intercept (b) by substituting the values of a point (x, y) and the slope (m) into the slope-intercept form:
2 = -2(5) + b
2 = -10 + b
b = 12
Therefore, the equation of the line passing through (5,2) and (4,4) is y = -2x + 12.
Step 3: Find the coordinates of the foot of the perpendicular.
To find the foot of the perpendicular, we need to find the intersection point of the line passing through (5,2) and (4,4) and the line perpendicular to it passing through (2,1).
The slope of a line perpendicular to another line is the negative reciprocal of the slope of the original line. Therefore, the slope of the perpendicular line is 1/2.
Using the point-slope form of a line, we can write the equation of the perpendicular line passing through (2,1):
y - 1 = (1/2)(x - 2)
2y - 2 = x - 2
x - 2y = 0
Now, let's solve the system of equations formed by the two lines:
y = -2x + 12
x - 2y = 0
Substituting the value of y from the first equation into the second equation, we get:
x - 2(-2x + 12) = 0
x + 4x - 24 = 0
5x = 24
x = 24/5
Substituting the value of x into the first equation, we get:
y = -2(24/5) + 12
y = -48/5 + 60/5
y = 12/5
Therefore, the coordinates of the foot of the perpendicular are (24/5, 12/5).
Step 4: Calculate the distance between the first vertex and the foot of the perpendicular.
Using the distance formula, we can find the distance between (2,1) and (24/5, 12/5):
d = √[(x2 - x1)^2 + (y2 - y1)^2]
= √[(24/5 - 2)^2 + (12/5 - 1)^2]
= √[(24/5 - 10/5)^2 + (12/5 -
Step 1: Plot the given vertices on a coordinate plane.
The given vertices of the triangle are (2,1), (5,2), and (4,4). Let's plot these points on a coordinate plane.
Step 2: Find the equation of the line passing through the second and third vertices.
To find the equation of the line passing through (5,2) and (4,4), we can use the slope-intercept form of a line, which is y = mx + b.
First, let's find the slope (m):
m = (y2 - y1) / (x2 - x1)
= (4 - 2) / (4 - 5)
= 2 / -1
= -2
Now, let's find the y-intercept (b) by substituting the values of a point (x, y) and the slope (m) into the slope-intercept form:
2 = -2(5) + b
2 = -10 + b
b = 12
Therefore, the equation of the line passing through (5,2) and (4,4) is y = -2x + 12.
Step 3: Find the coordinates of the foot of the perpendicular.
To find the foot of the perpendicular, we need to find the intersection point of the line passing through (5,2) and (4,4) and the line perpendicular to it passing through (2,1).
The slope of a line perpendicular to another line is the negative reciprocal of the slope of the original line. Therefore, the slope of the perpendicular line is 1/2.
Using the point-slope form of a line, we can write the equation of the perpendicular line passing through (2,1):
y - 1 = (1/2)(x - 2)
2y - 2 = x - 2
x - 2y = 0
Now, let's solve the system of equations formed by the two lines:
y = -2x + 12
x - 2y = 0
Substituting the value of y from the first equation into the second equation, we get:
x - 2(-2x + 12) = 0
x + 4x - 24 = 0
5x = 24
x = 24/5
Substituting the value of x into the first equation, we get:
y = -2(24/5) + 12
y = -48/5 + 60/5
y = 12/5
Therefore, the coordinates of the foot of the perpendicular are (24/5, 12/5).
Step 4: Calculate the distance between the first vertex and the foot of the perpendicular.
Using the distance formula, we can find the distance between (2,1) and (24/5, 12/5):
d = √[(x2 - x1)^2 + (y2 - y1)^2]
= √[(24/5 - 2)^2 + (12/5 - 1)^2]
= √[(24/5 - 10/5)^2 + (12/5 -
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The vertices of a triangle are (2,1),(5,2)and(4,4).find the length of the perpendicular from the first vertex to the opposite side.?
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The vertices of a triangle are (2,1),(5,2)and(4,4).find the length of the perpendicular from the first vertex to the opposite side.? for Class 12 2024 is part of Class 12 preparation. The Question and answers have been prepared according to the Class 12 exam syllabus. Information about The vertices of a triangle are (2,1),(5,2)and(4,4).find the length of the perpendicular from the first vertex to the opposite side.? covers all topics & solutions for Class 12 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The vertices of a triangle are (2,1),(5,2)and(4,4).find the length of the perpendicular from the first vertex to the opposite side.?.
The vertices of a triangle are (2,1),(5,2)and(4,4).find the length of the perpendicular from the first vertex to the opposite side.? for Class 12 2024 is part of Class 12 preparation. The Question and answers have been prepared according to the Class 12 exam syllabus. Information about The vertices of a triangle are (2,1),(5,2)and(4,4).find the length of the perpendicular from the first vertex to the opposite side.? covers all topics & solutions for Class 12 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The vertices of a triangle are (2,1),(5,2)and(4,4).find the length of the perpendicular from the first vertex to the opposite side.?.
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