The perpendicular bisector of the line segment joining the points \( A(1,5) \) and B \( (4,6) \) cuts the \( y \)-axis at
(A) \( (0,13) \)
(B) \( (0,-13) \)
(C) \( (0,12) \)
(D) \( (13,0) \)
Given:
The line segment joining the points \( A(1,5) \) and \( B(4,6) \).
To do:
We have to find the point at which the perpendicular bisector of the line segment joining the points \( A(1,5) \) and B \( (4,6) \) cuts the \( y \)-axis.
Solution:
We know that,
The perpendicular bisector of a line segment divides the line segment into two equal parts.
The perpendicular bisector of the line segment passes through the mid-point of the line segment.
Let the perpendicular bisector of $A B$ meets $\mathrm{y}$ axis at $\mathrm{P}(0, \mathrm{y})$
Therefore,
$\mathrm{AP}=\mathrm{BP}$
Squaring both sides, we get,
$AP^{2}=BP^{2}$
Using distance formula, we get,
$(\mathrm{x}_{1}-0)^{2}+(\mathrm{y}_{1}-\mathrm{y})^{2}=(\mathrm{x}_{2}-0)^{2}+(\mathrm{x}_{2}-\mathrm{y})^{2}$
$(1-0)^{2}+(5-\mathrm{y})^{2}=(4-0)^{2}+(6-\mathrm{y})^{2}$
$1+5^2-2(5)(y)+y^2=16+6^2-2(6)(y)+y^2$
$1+25-10y=16+36-12y$
$12y-10y=52-26$
$2y=26$
$y=13$
Therefore, the point is \( (0,13) \).
Related Articles
- The point which lies on the perpendicular bisector of the line segment joining the points \( A(-2,-5) \) and \( B(2,5) \) is(A) \( (0,0) \)(B) \( (0,2) \)(C) \( (2,0) \)(D) \( (-2,0) \)
- Find the equation of the perpendicular bisector of the line segment joining points $(7, 1)$ and $(3, 5)$.
- In what ratio does the point P $(-4,6)$ divide the line segment joining the points A$(-6,10)$ and B$(3,8)$ ?
- State whether the following statements are true or false. Justify your answer.Point \( P(0,2) \) is the point of intersection of \( y \)-axis and perpendicular bisector of line segment joining the points \( A(-1,1) \) and \( B(3,3) \).
- The mid-point $P$ of the line segment joining the points $A (-10, 4)$ and $B (-2, 0)$ lies on the line segment joining the points $C (-9, -4)$ and $D (-4, y)$. Find the ratio in which $P$ divides $CD$. Also, find the value of $y$.
- Find the coordinates of the point \( Q \) on the \( x \)-axis which lies on the perpendicular bisector of the line segment joining the points \( A(-5,-2) \) and \( B(4,-2) \). Name the type of triangle formed by the points \( Q, A \) and \( B \).
- Determine the ratio, in which the line $2x + y - 4 = 0$ divides the line segment joining the points $A(2, -2)$ and $B(3, 7)$.
- If $R (x, y)$ is a point on the line segment joining the points $P (a, b)$ and $Q (b, a)$, then prove that $x + y = a + b$.
- If $R\ ( x,\ y)$ is a point on the line segment joining the points $P\ ( a,\ b)$ and $Q\ ( b,\ a)$, then prove that $a+b=x+y$
- Find the mid point of the line segment joining the points $( 0,\ 0)$ and $( -2,\ -4)$.
- Find the mid point of the line segment joining the points $( 0,\ 0)$ and $( 2,\ 2)$.
- Whether the following statement is true or false. Justify your answer.Point A(2,7) lies on the perpendicular bisector of the line segment joining the points P(6,5) and Q$(0, -4)$.
- Find the mid-point of the line segment joining the points $A ( -2,\ 8)$ and $B ( -6,\ -4)$.
- Find the ratio in which the points $(2, y)$ divides the line segment joining the points $A (-2, 2)$ and $B (3, 7)$. Also, find the value of $y$.
- Draw a line segment $AB$ of length $5.8\ cm$. Draw the perpendicular bisector of this line segment.
Kickstart Your Career
Get certified by completing the course
Get Started