SOLUTION: The line y = x + 2 intersects the circle x2 + y2 = 10 in two points. Call the third- quadrant point R and the first-quadrant point E, and find their coordinates. Let D be the point

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Question 1154497: The line y = x + 2 intersects the circle x2 + y2 = 10 in two points. Call the third- quadrant point R and the first-quadrant point E, and find their coordinates. Let D be the point where the line through R and the center of the circle intersects the circle again. The chord DR is an example of a diameter. Show that triangle RED is a right triangle.
Found 2 solutions by Edwin McCravy, greenestamps:
Answer by Edwin McCravy(20060)   (Show Source): You can put this solution on YOUR website!
The line y = x + 2 intersects the circle x2 + y2 = 10 in two points. Call the
third- quadrant point R and the first-quadrant point E, and find their
coordinates. Let D be the point where the line through R and the center of the
circle intersects the circle again. The chord DR is an example of a diameter.
Show that triangle RED is a right triangle.



Find the coordinates of the points of intersection E and R:



We use the slope formula to find the slope of RE:



By symmetry, since R=(-3,-1), we can tell that D=(3,1)

We also use the slope formula to find the slope of ED:



So RE and ED are perpendicular because their slopes are negative
reciprocals of each other.  So RED is a right triangle.

Edwin
Answer by greenestamps(13203)   (Show Source): You can put this solution on YOUR website!


A rough sketch of the circle and line give a quick picture that makes it easy to solve the problem.

You could easily find the points of intersection algebraically by substituting y=x+2 in the equation for the circle.

However, a bit of simple mental arithmetic shows that the values of x and y, ignoring the signs, are 3 and 1; the rough sketch then determines that R is (-3,-1) and E is (1,3).

Then, since RD is a diameter (cutting the circle into two 180-degree arcs), angle RED is 90 degrees.
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