1)Find the distance between (3,4) and the line 4x + 3y + 7 = 0.
2)Find the distance between the graphs of 5x - 12y + 45 = 0 and 5x - 12y - 46 = 0.
3)A circle centered at the origin is tangent to the line x - y + 4 = 0. What is the area of the circle?
4)Find the area of the region bounded by the lines y = 5 and 5x + 2y = 25 and the coordinate axes.
5)We are given A = (2,5), B = (-5, 2), C = (-2, -5), and D = (5,-2). Find [ABCD].
6)For some positive real number r, the line x + y = r is tangent to the circle x^2 + y^2 = r. Find r.
7)We are given O = (0,0), A = (5,2) and B = (6, -3). Find [OAB].
8)A circle centered at (3,2) is tangent to the line y = (x+1)/3. Find the area of the circle.
9)A line with slope 3 is 2 units away from the origin. Find the area of the triangle formed by this line and the coordinate axes.
10)The line y = (3x + 20)/4 intersects a circle centered at the origin at A and B. We know the length of chord AB is 20. Find the area of the circle.
11)The lines y = (5/12)x and y = (4/3)x are drawn in the coordinate plane. Find the slope of the line that bisects the acute angle between these lines.
Picture: https://latex.artofproblemsolving.com/4/b/c/4bc61e6a0f1f043a88e9efa268018820d89cea3a.png
1)Find the distance between (3,4) and the line 4x + 3y + 7 = 0
We can use this to find the answer.....the distance between (a, b) and the line AX + BY + C = 0 is given by
l A(a) + B(b) + C l
_________________
√ [ A^2 + B^2 ]
So we have
l 4(3) + 3(4) l l 24 l 24
___________ = _______ = ___ = 4.8 units
√ [ 4^2 + 3^2] √25 5
2)Find the distance between the graphs of 5x - 12y + 45 = 0 and 5x - 12y - 46 = 0.
The first line has a y intercept of
-12 y = -45
y = 45 / 12
y = 15/4 = 3.75
So....the point (0, 3.75) is on the first line
And the distance betwen this point and the second line will be the perpendicular distance between the two lines....so...we have
l 5 (0) - 12 (3.75) - 46 l l 91 l 91
____________________ = ___________ = ___ = 7 units
√[ 5^2 + 12^2] √169 13
3)A circle centered at the origin is tangent to the line x - y + 4 = 0. What is the area of the circle?
The distance from the centrer to the line will be the radius of the circle....so we have
l (0) - (0) + 4 l 4
______________ = ____ = 2√2 = √8 = the radius
√[ 1^2 + 1^2] √2
So...the area of the circle is
pi * ( √8)^2 =
8pi units^2
4)Find the area of the region bounded by the lines y = 5 and 5x + 2y = 25 and the coordinate axes.
Look at the following pic:
This are will be the difference in the area of triangle ADB - area of triangle ACE
The area of triangle ADB = (1/2) base (height) = (1/2) (5)(12.5) = 31.25 units^2
Since the triangles are similiar, the linear scale factor of ACE to ADB = 3/5
So...the area of triangle ACE = area of ADB * (3/5)^2 = 31.25 * (3/5)^2 = 11.25 units^2
So...the area of the bounded region is 31.25 - 11.25 = 20 units^2
5)We are given A = (2,5), B = (-5, 2), C = (-2, -5), and D = (5,-2). Find [ABCD].
We can split this up into the area of two triangles.....ABD and CBD
BD will serve as the base for both
And its length is √ [ (-5-5)^2 + (-2 -2)^2 ] =√[ 10^2 + 4^2 ] = √116 = 2√29 units
And the slope of BD is [ -2 - 2 ] / [ 5 - -5] = [-4] / [10] = -2/5
And the equation of the line containing BD is
y = (-2/5)(x -5) - 2
5y = -2(x - 5) - 10
5y = -2x + 10 - 10
5y + 2x = 0
2x + 5y = 0
So...the distance from A to this line will be the height of triangle ABD =
l 2(2) + 5(5) l 29
___________ = ____ = √29 units
√ [2^2 + 5^2 ] √29
So...the area of triangle ADB = (1/2) *2√29 *√29 = 29 units^2
And the distance between C and this line will be the height of triangle CBD
l 2(-2) + 5(-5) l 29
_____________ = _____ = √29 units
√29 √29
And triangle CBD will have the same area as triangle ABD
So...the total area of [ABCD ] = 58 units^2
6)For some positive real number r, the line x + y = r is tangent to the circle x^2 + y^2 = r. Find r.
At the point of tangency, x = x^2 and y = y^2
And these will be true at 0 and 1
But...x, y cannot = 0 since the center of the circle is (0,0)
So...x, y = 1
And
x + y =
1 + 1 =
2 = r
7)We are given O = (0,0), A = (5,2) and B = (6, -3). Find [OAB].
We have triangle OAB
Let AB be the base
And its length is
√ [ (6 - 5)^2 + (-3 - 2)^2 ] = √ [ 1^2 + 5^2 ] = √26
And the slope between these two points is [-3 -2 ] / [ 6 -5] = --5
And the equation of the line containing AB is
y =-5(x - 5) + 2
y = -5x + 25 + 2
y = -5x + 27
5x + 1y -27 = 0
And the distance from (0,0) to this line forms the height of OAB
l 5(0) + 1(0) -27] 27
_______________ = ____
√[ 5^2 + 1^2 ] √26
So....the area of OAB is (1/2)√26 * 27/√26 = 13.5 units^2
I worked (9) but my answer disappeared...it is a little lengthy to re-create...but....the answer ends up as 20/3 units^2
8)A circle centered at (3,2) is tangent to the line y = (x+1)/3. Find the area of the circle.
y = (x + 1) / 3 can be written as
y = (1/3)x + 1/3
3y = x + 1
x - 3y + 1 = 0
The distance from ( 3,2) to this line will be the radius of the circle
l 3 - 3(2) + 1 l 2 √10 √2
____________ = ____ = ___ = ___
√[ 1^2 + 3^2 ] √10 5 √5
So...the area of the circle is
pi * ( √ (2/5) ) ^2 =
2pi / 5 units^2
10)The line y = (3x + 20)/4 intersects a circle centered at the origin at A and B. We know the length of chord AB is 20. Find the area of the circle.
We can write the line as
4y = 3x + 20
3x - 4y + 20 = 0
The perpendicular distance between the center of the circle and the line will bisect the chord
And this distance is
l 3(0) - 4(0) + 20 l 20
________________ = ______ = 4
√[3^2 + 4^2] = 5
A right triangle will be formed by the bisected chord, the perpendicular distance from the center to the midpoint of the chord and the radius which will form the hypotenuse.
So....the radius is √[10^2 + 4^2 ] = √116 units
And the area of the circle is
pi * ( √116)^2 =
116 pi units^2
Why in the world are you posting copyrighted problems? It's illegal, and it also deprives the whole point of the Geometry class. The whole point of the homework is to practice the problems so you can study for the midterm, and asking others to do it for you won't help at all. Besides, the AoPS Admins already know you are cheating, so doing this won't help at all.
11)The lines y = (5/12)x and y = (4/3)x are drawn in the coordinate plane. Find the slope of the line that bisects the acute angle between these lines.
Construct a circle centered at the origin with a radius of 1
The equation is
x^2 + y^2 = 1
We can find the x coordinate of the intersection of the first line and this circle,thusly "
x^2 + [ (5/12)x ]^2 = 1
x^2 + (25/144)x^2 =1
[144 + 25] / 144 x^2 = 1
[169] /144 x^2 =1
x^2 = 144/169
x = 12/13
And y = (5/12)(12/13) = 5/13
So ( 12/13, 5/13 ) is on the circle
Like wise...we can find the intersection of this circle with the second line :
x^2 +[ (4/3)x\^2 =1
x^2 + (16/9)x^2 = 1
[9 + 16 ] / 9 * x^2 =1
25/9 x^2 =1
x^2 = 9/25
x = 3/5
And y = (4/3)(3/5) = 4/5
So (3/5, 4/5) is on the circle
And a chord can be drawn on this circle with endpoints ( 12/13, 5/13) and (3/5, 4/5 )
And the midpoint of this chord is
( [12/13 + 3/5] / 2, [ 5/13 + 4/5 ] / 2 ) =
(99/130 , 77/130 )
And the line drawn from the origin to this point is the line we seek and it will have the slope
[77/130] / [99/130] = 77/99 = 7/9