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Question 554469: hi, i need your help please... i don't know how to answer these questions because i can't fully imagine them... i hope you can help me because i really want to understand this lesson for the sake of my grades.....
a) the sum of the distance from a point P to (4,0) and (-4,0) is 9. if the abscissa of P is 1, find its ordinate...
b)the center of a circle is at (-3,-2). if a chord of length 4 is bisected at (3,1), find the length of the radius...
THANK YOU!!!!!
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! to begin with, we need to understand what ordinate and abscissa mean.
the following link addresses that issue:
http://www.blurtit.com/q550444.html
the short answer is:
abscissa is the x value of the coordinate.
ordinate is the y value of the coordinate.
they are giving you the x value of the point P.
that's the abscissa of 1.
you have to find the ordinate of the point P.
that would be the y value of the point P.
coordinates are given in pairs of (x,y).
x is the first value of the pair and points to the x-value of the point.
y is the second value of the pair and points to the y-value of the point.
a diagram of what i think your triangle will look like is shown below:
the triangle is labeled APC.
there is a perpendicular from point P to point D on AC.
point A is (-4,0)
point P is (1,y) ***** y is the value we are looking for.
point C is (4,0)
point D is (1,0)
the abscissa of point P is 1 which makes it the x value of the coordinate pair represented by point P.
you are looking for the ordinate of point P.
the ordinate is the y value of the coordinate pair.
i labeled it "y" until we could find a value for it.
you are given that the distance between point A and C is 8.
you are also given that the abscissa of the point P is 1.
this means that the vertical line to point P lies on the value of x = 1.
this means that there are 5 units from point A to D.
this means that there are 3 units from point D to C.
you need to find the length of the line from point P to D.
this is the altitude of the right triangles formed.
the 2 right triangles formed are APD and CPD.
all you know is that the sum of the distance from point P to A and P to C is equal to 9.
in the diagram i labeled the following:
the length of the line AP is called b.
the length of the line PC is called a.
the length of the line PD is called c.
by using the pythagorean formula, we can find the value of c.
the pythagorean formula states that the hypotenuse squared is equal to the sum of the legs squared.
in triangle APD, b is the hypotenuse and 5 and c are the legs.
in triangle CPD, a is the hypotenuse and 3 and c are the legs.
our formulas that we start with are:
5^2 + c^2 = b^2
3^2 + c^2 = a^2
we solve for c^2 in both formulas to get:
c^2 = b^2 - 5^2
c^2 = a^2 - 3^2
since they are both equal to c^2, then they are both equal to each other, so we get:
b^2 - 5^2 = a^2 - 3^2
we are given that a + b = 9
from that we can solve for b in terms of a to get:
b = 9 - a
we can then substitute for b in the equation of b^2 - 5^2 = a^2 - 3^2 to get:
(9 - a)^2 - 5^2 = a^2 - 3^2
we now solve for a.
we simplify the equation to get:
81 - 18a + a^2 - 5^2 = a^2 - 3^2
we subtract a^2 from both sides of the equation and we add 5^2 to both sides of the equation to get:
81 - 18a + a^2 - a^2 - 5^2 + 5^2 = a^2 - a^2 - 3^2 + 5^2
we combine like terms to get:
81 - 18a = -3^2 + 5^2
we simplify further to get:
81 - 18a = -9 + 25
we combine like terms again to get:
81 - 18a = 16
we subtract 81 from both sides of this equation to get:
- 18a = 16 - 81
we combine like terms to get:
- 18a = - 65
we divide both sides of this equation by -18 to get:
a = 65/18 which simplifies to:
a = 3.611111111
sinca a + b = 9, this means that:
a = 3.611111111
b = 5.388888889
we now have a value for a and b and we can use those values to solve for c using the pythagorean formula again.
the result of that operation is that:
c = 2.010005835
we rounde all answers to the nearest hundredth or whatever rounding is required to get:
a = 3.61
b = 5.39
c = 2.01
the value of c is the length of the line PD which also becomes the y value of the coordinate of P.
the coordinate of P becomes (1,2.01)
the diagram of your triangles formed is shown below:
--------------------------------
your second problem starts here
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the center of your circle is at (-3,-2)
your chord is bisected at (3,1)
the length of chord is 2.
you want to find the length of the radius of your circle.
the attached diagram shows what i think is the relationship you are looking for.
the radius of a circle can form the perpendicular bisector of any chord in that circle.
this is one of the theorems in geometry that you should probably have already studied.
this is because the radii of the circle form an isosceles triangle with the chord as shown in the diagram.
that isosceles triangle is ABC in the diagram.
it is an isosceles triangle because the sides of the triangle are the radii of the circle.
since you have an isosceles triangle, then the altitude of that triangle is the perpendicular bisector of the base of that triangle which is the chord.
in your problem, the center of the circle is at (-3,-2) and the bisecting point on the chord is (3,1).
this means that you can draw a line from the center of the circle to that point and the line formed is the perpendicular bisector of the triangle.
that perpendicular bisector forms 2 right triangles which are triangle ADB and ADC.
the base of each of those triangle is 2.
the length of the perpendicular bisector which is the line AD is given by the equation:
AD = sqrt (x^2 + y^2) where x and y are the coordinate points of point A and point D.
the formula becomes:
AD = sqrt (-3 - 3)^2 + -2 - 2)^2 which becomes:
AD = sqrt ((-6)^2 + (-4)^2) which becomes:
AD = sqrt (9 + 16) which becomes:
AD = sqrt(25) which becomes:
AD = 5
the length of AD is equal to 5 as shown in the diagram.
the length of AB is found using the pythagorean formula of:
(AB)^2 = 5^2 + 2^2 which becomes:
(AB)^2 = 25 + 4 which becomes:
(AB)^2 = 29
take square root of both sides of this equation to get:
AB = sqrt(29).
that's your answer.
if you solve for AC, you will find that it's value is also sqrt(29).
diagram for your second problem is shown below:
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