Find the equation of the circle incribed in a triangle, if the angle has its sides on the lines:
2x + y - 9 = 0, -2x + y - 1 = 0, and -x + 2y +7 = 0. Draw the figure.
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To find the equation of the inscribed circle in a triangle, we need to determine the coordinates of the incenter (the center of the circle) and the radius of the circle.
Step 1: Find the intersection point of the given three lines to determine the incenter.
To find the intersection point, we can solve the system of equations formed by the three lines.
2x + y - 9 = 0
-2x + y - 1 = 0
-x + 2y + 7 = 0
To solve this system, we can use various methods like substitution or elimination. Here, we'll use the elimination method.
Adding the first two equations, we get:
(2x + y - 9) + (-2x + y - 1) = 0
2y - 10 = 0
2y = 10
y = 5
Substituting y = 5 into the first equation:
2x + 5 - 9 = 0
2x - 4 = 0
2x = 4
x = 2
So, the incenter coordinates are (2, 5).
Step 2: Find the lengths of the sides of the triangle.
We have the equations of the sides of the triangle:
2x + y - 9 = 0
-2x + y - 1 = 0
-x + 2y + 7 = 0
The lengths of the sides of the triangle can be found using the distance formula:
d = squareroot((x2 - x1)^2 + (y2 - y1)^2)
Using the coordinates of the intersection point (2, 5), we can calculate the lengths of the three sides.
For the side with equation 2x + y - 9 = 0:
d1 = squareroot((2 - 2)^2 + (5 - 5)^2) = 0
For the side with equation -2x + y - 1 = 0:
d2 = squareroot((2 - (-2))^2 + (5 - 5)^2) = 4
For the side with equation -x + 2y + 7 = 0:
d3 = squareroot((2 - (-3))^2 + (5 - 10)^2) = squareroot(25 + 25) = 5
Step 3: Calculate the inradius.
The inradius of a triangle is the radius of the inscribed circle. It can be found using the formula:
inradius = Area / Semi-perimeter
To find the area of the triangle, we can use Heron's formula:
Area = squareroot(s * (s - d1) * (s - d2) * (s - d3))
where s is the semiperimeter.
The semiperimeter can be calculated as:
semiperimeter = (d1 + d2 + d3) / 2
Using the values calculated earlier, we find:
semiperimeter = (0 + 4 + 5) / 2 = 9/2 = 4.5
Now, we can calculate the area:
Area = squareroot(4.5 * (4.5 - 0) * (4.5 - 4) * (4.5 - 5))
= squareroot(4.5 * 0.5 * 0.5 * (-0.5))
= squareroot(0.5625)
≈ 0.75
Finally, we can calculate the inradius:
inradius = Area / Semiperimeter
= 0.75 / 4.5
≈ 0.1667
So, the equation of the inscribed circle in the triangle is:
(x - 2)^2 + (y - 5)^2 = 0.1667^2
Now that you have the equation of the inscribed circle, you can draw the figure by plotting the incenter (2, 5) and drawing the circle with a radius of approximately 0.1667 units.