Find the coordinates of a point C from the given line, point and circle

• Mathman2013
In summary, using Geogebra and analytic geometry, the acute angle between the point P(2,8) and the line m: y = -0.75*x+3.25 can be found to be 45.58 degrees. The coordinates of point B are (-4.92,6.94) and the point C can be found by calculating the perpendicular distance from P to the line and using the area of the triangle to find the base, which is the distance BC. Another way to find the point C is by finding the intersection of a perpendicular line through P and the line m. The distance between points P and E is 5 units and the angle of interest is 45.58 degrees.
Mathman2013
Homework Statement
Let the point P(2,8) be a point in xy-plane and line m: y = -0.75*x+3.25 be a line in the xy-plan. Using CAS, find The distance from a point P to a point B is 7 unites. Where the x coordinate of B is negative. Find the acute angle between PB and m. The distance PM, PC and PB form a triangle of area 20. The area of the formed triangle is 20. Find the coordinates of C.
Relevant Equations
y= mx+b and (x-a)^2+(y-b)^2=r^2
Let the point P(2,8) be a point in xy-plane and line m: y = -0.75*x+3.25 be a line in the xy-plan. The distance from a point P to a point B is 7 unites. Where the x coordinate of B is negative. Find the acute angle between PB and m.

To find B I then construct a circle of radius 7 with center C(2,8), and find that the coordinates of B to be (-4.92;6.94).

I use Geogebra, to find the acute angle to be 45.58 degrees.

My question if I need to find the coordinat C. Isn't the info that the area of the triangle is 20 redundant? Because I see from Geogebra that the coordinates for C must be (2.92; 1.06)

Why do you assume that C lies on the circle? I don't think it says that anywhere.

Mathman2013
phyzguy said:
Why do you assume that C lies on the circle? I don't think it says that anywhere.
Thank you for your answer. I look at the text again and the point C is suppose to be on line l too.

So my question how or can I use the information that the area of the triangle is suppose to be 20 to find C? Or is that information redundant?

So the point C is on the line, but not necessarily on the circle. So calculate the perpendicular distance from P to the line. This is the height of the triangle. Then the area of the triangle is Base*Height/2 = 20, so you can calculate the base, which is the distance BC. Since you know the coordinates of B, you should be able to find the point C.

Mathman2013
phyzguy said:
So the point C is on the line, but not necessarily on the circle. So calculate the perpendicular distance from P to the line. This is the height of the triangle. Then the area of the triangle is Base*Height/2 = 20, so you can calculate the base, which is the distance BC. Since you know the coordinates of B, you should be able to find the point C.
Just to understand your point. Since both B and C are on the l. Then the base of the triangle must be the distance from BC? If that that's the base. Then if I use the point-to-point distance formula, then I get a base of 9.8. And if the area of the triangle is 20 then the height is 9.8*height/2 = 20 then I get height = 4.08.

But the distance from P to the line l which should be the same as the height of the triangle 5? So what I am doing wrong?

I think you are still assuming that the point C is on the circle, but you don't know that.
I think you have correctly calculated the perpendicular distance from P to the line I as 5.0. This is the height. But you don't know the distance BC, since you don't know where the point C is. You have to find the point C so that the area of the triangle PBC is 20.

Mathman2013
I
phyzguy said:
I think you are still assuming that the point C is on the circle, but you don't know that.
I think you have correctly calculated the perpendicular distance from P to the line I as 5.0. This is the height. But you don't know the distance BC, since you don't know where the point C is. You have to find the point C so that the area of the triangle PBC is 20.
I think I got it now. if the height is 5 units = distance from line m to P. Then its is suppose to satisfy that (heigh*base)/2 = 20 meaning that (5*base)/2 = 20. Meaning that the base is suppose to be 8.

That must mean the distance from B to C is 8. Then I draw a circle with B as its center and radius 8, and find the intersection between this circle and m. which is the coordinate of C ?

Looks good to me !

There is another way to do it using analytic geometry and without using the information that the area of the triangle is 20.

Any line perpendicular to line m has the equation ##y=\dfrac{1}{0.75}x+q##. The one that goes through P(2,8) has intercept ##q## that is found from $$8=\frac{1}{0.75}2+q\implies q=\frac{16}{3}.$$Thus, the perpendicular line h in post #7 has the equation ##y=\dfrac{4}{3}x+\dfrac{16}{3}##.

We can now find the coordinates of the intersection E of lines m and h:
$$\frac{4}{3}x_E+\frac{16}{3}=-\frac{3}{4}x_E+\frac{13}{4}\implies x_E=-1\implies y_E=4.$$
The distance between points P and E is ##d=\sqrt{ (2-(-1))^2+(8-4)^2}=\sqrt{9+16}=5.##
The angle of interest is given by ##\alpha=\arcsin(5/7)=45.58^o##.

1. How do I find the coordinates of a point C from a given line and point?

To find the coordinates of a point C, you can use the distance formula to calculate the distance between the given point and the line. Then, using the slope of the line, you can determine the coordinates of C by adding or subtracting the distance from the given point.

2. Can I find the coordinates of point C from just a line and circle?

Yes, you can find the coordinates of point C by first finding the intersection point(s) between the line and the circle. Then, using the distance formula, you can determine the distance between the intersection point(s) and the center of the circle. Finally, using the Pythagorean theorem, you can calculate the coordinates of point C by adding or subtracting the distance from the intersection point(s) along the x and y axes.

3. Is it possible to find multiple coordinates for point C from a given line, point, and circle?

Yes, it is possible to find multiple coordinates for point C if the line intersects the circle at more than one point. In this case, there will be multiple intersection points, and each one can be used to find a different set of coordinates for point C using the distance and Pythagorean formulas.

4. What if the line is parallel to the circle? Can I still find the coordinates of point C?

If the line is parallel to the circle, there will be no intersection points and thus it is not possible to find the coordinates of point C. In this case, the line and circle do not intersect and there is no point C that lies on both the line and the circle.

5. Are there any other methods for finding the coordinates of point C from a given line, point, and circle?

Yes, there are other methods for finding the coordinates of point C, such as using the equations of the line and circle to solve for the coordinates algebraically. Additionally, you can use geometric constructions, such as constructing perpendicular lines or tangent lines, to determine the coordinates of point C.

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