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Question 772294: 1.Find the equation of the ciecle inscribed in a triangle, if the triangle has its sides on the line; 2x + y - 9 = 0, -2x + y - 1 = 0, and -x + 2y + 7 = 0. Draw the figure.
2.identify the points of the ellipse 4x^2 + 5y^2 - 8x + 20y = -24.
3. Find the vertices, foci, eccentricity, and lenght of the latus rectum of the ellipse x^2 + 16y^2 = 16.
Answer by solver91311(24713)   (Show Source):
You can put this solution on YOUR website!
The perpendicular bisectors of the three sides of any triangle intersect in a point called the circumcenter which is a point equidistant from each of the sides and is therefore the center of the inscribed circle; said circle having a radius equal to the distance from the circumcenter to any one of the sides.
Step 1: Find one vertex of the triangle by solving the 2X2 linear system comprised of two of the equations of lines containing segments representing the sides of the triangle.
Step 2: Repeat step 1 twice more so that you have solved for the intersection of each of the three pairs of sides thereby identifying all three vertices.
Step 3: Using the mid-point formulas and the coordinates of the three vertices, determine the mid points of two of the sides.
Step 4: Using the equation for the line containing one of the segments for which you determined a mid-point in step 3, calculate the slope of the line containing that side.
Step 5: Repeat step 4 for the other side for which you have calculated a mid-point.
Step 6: Using the point-slope form of an equation of a line, the slope you calculated in step 4, and the corresponding mid-point, derive the equation of the perpendicular bisector of that side.
Step 7: Repeat step 6 for the other side for which you have a slope and a midpoint.
Step 8: Solve the 2X2 system formed by the equations derived in steps 6 and 7.
Step 9: The solution from step 8 is the ordered pair representing the center of the inscribed circle. Using that center point and either of the midpoints calculated in step 3 in the distance formula, calculate the radius of the inscribed circle. For convenience, leave the result in  form.
Step 10: Using the coordinates of the center from step 8 and the value of from step 9, write the equation of the inscribed circle:
^2\ +\ (y\ -\ k)^2\ =\ r^2)
Where ) is the center of the circle and  is the radius.
This problem requires that you take it slowly, checking your work after each step. One little sign error in an early step will turn into hours of drugery.
By the way, if you are wondering what happened to the other two problems you posted, go back and re-read the instruction that says "One question per post". Then you can stop wondering, right?
John
Egw to Beta kai to Sigma
My calculator said it, I believe it, that settles it
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