Show that #CM# and #RQ# are perpendicular ?

Question

Wyatt Anderson · Accepted Answer

See below.

Given #R=(3a,a)# and #Q=(a,-3a)# the mid point of #RQ# is given by

#M=1/2(R+Q)=(2a,-a)#

The circle's center is #C=(0,0)#

so #CM=(2a,-a)-(0,0)=(2a,-a)# and

#RQ = R-Q =(2a,4a) #

Finally

#RQ cdot CM = (2a,4a) cdot (2a,-a) = 4a^2-4a^2=0#

Attached a figure

Claire Akin · Answer

The x coordinate of the chord goes from #3a# to #a#, therefore, the x coordinate of the midpoint is halfway between them, 2a.

The y coordinate of the chord goes from #a# to #-3a#, therefore, the y coordinate of the midpoint is halfway between them, -a.

The coordinates of the midpoint are #M(2a,-a)#

The slope, #m_1#, of the chord RQ is:

#m_1 = (-3a - a)/(a - 3a)#

#m_1 = (-4a)/( -2a)#

#m_1 = 2#

The equation of the circle #x^2 + y^2 = 10a^2# is a special case of the more general equation #(x - h)^2+(y-k)^2 = r^2# where the center is the origin #(0,0)#.

The slope, #m_2#, of the line from the center to the midpoint is:

#m_2 = (-a - 0)/(2a - 0)#

#m_2 = (-a)/(2a)#

#m_2 = -1/2#

Please observe that:

#m_1m_2 = -1#

This shows that the two lines are perpendicular.

Claire Akin · Answer

undefined To show that CM and RQ are perpendicular, we can use the property of perpendicular lines in coordinate geometry. If the product of the slopes of two lines is -1, then they are perpendicular. Let's denote C as (x1, y1), M as (x2, y2), R as (x3, y3), and Q as (x4, y4).

The slope of line CM, m_CM, is (y2 - y1) / (x2 - x1).
The slope of line RQ, m_RQ, is (y4 - y3) / (x4 - x3).

If CM and RQ are perpendicular, then:
m_CM * m_RQ = -1.

So, we need to show that:
((y2 - y1) / (x2 - x1)) * ((y4 - y3) / (x4 - x3)) = -1.

If this equation holds true, then CM and RQ are perpendicular.

Sophia Alfred · Answer

undefined To show that CM and RQ are perpendicular, we can use the concept of slopes. Perpendicular lines have slopes that are negative reciprocals of each other.

Let's denote CM as line m and RQ as line n.

The slope of CM (m) can be calculated using the coordinates of points C and M. Similarly, the slope of RQ (n) can be calculated using the coordinates of points R and Q.

If the product of the slopes of two lines is -1, then the lines are perpendicular.

Therefore, to demonstrate that CM and RQ are perpendicular, we need to show that the product of their slopes is -1.