SOLUTION: Let A(-2,4), B(-7,1) and C(-1,-5) be the vertices of a triangle. a. Find an equation whose graph is a line that contains the median from A to the midpoint of line segment BC. b.

Algebra ->  Linear-equations -> SOLUTION: Let A(-2,4), B(-7,1) and C(-1,-5) be the vertices of a triangle. a. Find an equation whose graph is a line that contains the median from A to the midpoint of line segment BC. b.       Log On


   

Question 198824: Let A(-2,4), B(-7,1) and C(-1,-5) be the vertices of a triangle.
a. Find an equation whose graph is a line that contains the median from A to the midpoint of line segment BC.
b. Find the length of the median from A to line segment BC.
c. Find the length of the altitude from A to line segment BC.
d. Find the area of the traingle ABC.
Thanks.
Answer by solver91311(24713) About Me  (Show Source):
You can put this solution on YOUR website!

Let A(-2,4), B(-7,1) and C(-1,-5) be the vertices of a triangle.

a. Find an equation whose graph is a line that contains the median from A to the midpoint of line segment BC.

Use the midpoint formulas to find the midpoint of the segment BC.

and



where and are the coordinates of points B and C. (doesn't matter which is which)

Then use the two-point form of the equation of a line to derive the desired equation:



Where are the coordinates of point A and are the coordinates of the midpoint derived in the previous step.

b. Find the length of the median from A to line segment BC.

Use the distance formula:



Where are the coordinates of point A and are the coordinates of the midpoint derived in the previous step.

c. Find the length of the altitude from A to line segment BC.

Step 1. Derive the equation of the line containing segment BC. Use the two-point form of the equation of a line:



where and are the coordinates of points B and C. Put the equation into slope-intercept form (

Step 2. Determine the slope of the line that contains the altitude segment, namely a line perpendicular to segment BC passing through A. First you need the slope of the line containing BC which you can obtain by inspection of the slope-intercept form of the equation of the line containing BC derived in Step 1.

The slope of the line containing the altitude segment is the negative reciprocal of the slope of the line containing BC because:



Step 3. Derive the equation of the line containing the altitude segment. Use the point-slope form of the equation of a line:



Where are the coordinates of point A and is the slope number calculated in step 2. Put this equation in slope-intercept form as well.

Step 4. Solve the system of equations derived in steps 1 and 3 to determine the point of intersection between the altitude and segment BC. Since both equations are in slope intercept form, you can simply equate the two right-hand sides and solve for , then substitute back into either equation to calculate . This is where this problem gets really ugly -- for example the -coordinate of the point of intersection is -- like a mud fence. But persevere and you will be rewarded in the end.

Step 5. Use the distance formula:



Where and are the coordinates of point A and the point of intersection calculated in step 4.

d. Find the area of the traingle ABC.

Use the distance formula to calculate the measure of segment BC:



Where and are the coordinates of points B and C. Call this the measure of the base of the triangle.

Use the area of a triangle formula:



where is the measure of the base and is the measure of the altitude calculated in part c of the question.

John