consider the following statement: if a circle with its center at the origin has a chord with slope m, the equation of the right bisector of the chord is y=mx. Is this statement true or false? explain

false.

consider a horizontal chord, with slope=0

the right bisector is vertical, with undefined slope, not y=0x

This statement is false.

A right bisector of a chord divides the chord into two equal parts and intersects it at a 90-degree angle. In the case of a circle with its center at the origin, the equation of the chord passing through the origin would be in the form of y = mx, where m represents the slope of the chord.

However, the equation of the right bisector of the chord would not be y = mx. The slope of the right bisector would be the negative reciprocal of the original chord's slope. So, the correct equation of the right bisector would be y = -1/mx. This is because the perpendicular line has a slope that is the negative reciprocal of the original line's slope.

Therefore, the correct statement should be: if a circle with its center at the origin has a chord with the slope m, the equation of the right bisector of the chord is y = -1/mx.

To determine whether the given statement is true or false, we need to analyze the properties of chords and right bisectors in a circle.

Let's assume we have a circle with its center at the origin (0, 0) and a chord with a slope of m. The equation of a line can be represented in the form y = mx + b, where m is the slope and b is the y-intercept.

First, let's find the equation of the chord. Since the circle's center is at the origin, the equation of any line passing through the origin can be written as y = mx.

Now, let's consider the equation of the right bisector of the chord. The right bisector of a line segment is a line that intersects the line segment at a 90-degree angle, dividing it into two equal parts.

For any line segment, the slope of the right bisector is the negative reciprocal of the slope of the segment. In this case, the negative reciprocal of m is -1/m.

Therefore, the equation of the right bisector should be y = -1/m * x. Hence, the given statement is false.

The correct equation of the right bisector of the chord in this scenario would be y = -x/m, not y = mx.