Perpendicular Bisectors 1
Student Quiz - Perpendicular bisectors 1
Use this 11 question quiz for some quick fire practise and revision on the topic of perpendicular bisectors. The questions will ask you about the null and alternate hypotheses, 1 and 2 tailed tests and ask you to calculate p-numbers and use them to conclude your test for different significance levels. Do these on the go on your mobile or at your desk with a pen and paper by your side to help you. Answer the questions as best you can and then see how you did and read the solution notes included. Make a note of anything you misunderstood and revisit this quiz and many others as often as you like!
A line segment has a gradient of \(3\). What is the gradient of the perpendiular line?
If the gradient is \(3\), we can write the gradient as \(\frac{3}{1}\) so that the perpendicular gradient is \(-\frac{1}{3}\).
Find the centre and radius of the largest empty circle in the Voronoi diagram below.
Since there are two vertices, we will need to check the distance between each vertex and a randomly picked site that each vertex seperates.
For the vertex \((9,5)\), if I randomly pick site A, the distance is given by, \(d=\sqrt{(9-13)^2+(5-4)^2}=\sqrt{(-4)^2+1^2}=\sqrt{17}\)
For the vertex (5, 6), if I randomly pick site D, the distance is given by,
\(d=\sqrt{(5-0)^2+(6-9)^2}=\sqrt{5^2+3^2}=\sqrt{34}\)
Therefore, the largest empty circle has centre \((5, 6)\) and radius \(\sqrt{34}\).
In the Voronoi diagram below, a new site D has been added.
Choose which statement below is the correct process to create a new Voronoi diagram for sites A, B, C and D.
When adding a new site, a perpendicular bistector needs to be found between te new site, and the site which already exists in the cell. Then a new perpendicular bistector must be found between the new site, and any sites in cells which are directly next to the new site.
The rainfall in three nearby cities, A, B, C and D, over the course of the previous month, is 15 cm, 22 cm, 18 cm and 20 cm respectively. Use nearest neighbour interpolation to estimate the rainfall at (2, 3).
cm
The point (2, 3) is located on an edge between the sites C and D. We therefore take the mean of the rainfalls at C and D: \(\frac{18+20}{2}=19\ cm\)
If the point was located inside a cell, then we would estimate the rainfall to be the same as the site within that cell.
A line segment has a gradient of \(3\). What is the gradient of the perpendicular line?
If the gradient is \(3\), we can write the gradient as \(\frac{3}{1}\) so that the perpendicular gradient is \(-\frac{1}{3}\).
What is the gradient of the perpendicular bisector of the points (1,1) and (7,3)?
The gradient between the two points is \(\frac{1}{3}\). Hence, the perpendicular gradient is \(-\frac{3}{1}=-3\).
The midpoint of a line segment AB is (5, -2), and the point A is given as (10, -1). Find point B, giving your answer as a co-ordinate.
To get from the midpoint to point A, we must go 5 left and 1 up. It follows that to get to point B we must go 5 left and 1 up from the midpoint.
If the gradient of the line segment AB is \(\frac{1}{3}\), then the gradient of the perpendicular bisector is \(\frac{-1}{\frac{1}{3}}=-3\). We notice that the perpendicular bisector passes through (0,-4) and therefore it has a y-intercept of -4 (i.e. c = -4).
The four points shown below make a square. Find the midpoint of the diagonal BD.
Given that every length of the square is the same, the point B is (1,1) and point D is (4, 4). Using the midpoint formula, we find \(M=\left(\frac{1+4}{2},\ \frac{1+4}{2}\right)=\left(2.5,2.5\right)\).
Two points on a co-ordinate grid are given as X(-2, 8) and Y(4, -4). Find the perpendicular bisector of the points X and Y, giving your answer in the form y = mx + c.
To start, we can use the gradient formula from the formula booklet to find the gradient of the line connecting both points:
\(\frac{8--4}{-2-4}=\frac{12}{-6}=-2\)
Given that we want the equation of the perpendicular bisector, the perpendicular gradient is therefore \(\frac{-1}{-2}=\frac{1}{2}\).
The midpoint of the line XY is then found to be \(\left(1,2\right)\).
Finally, using \(y-y_{1}=m\left(x-x_{1}\right)\) from the formula booklet, we obtain:
\(y-2=\frac{1}{2}(x-1)\)
Thus, \(y=\frac{1}{2}x+\frac{3}{2}\)
Two points on a co-ordinate grid are given as X(1,1) and Y(4, 10). Find the midpoint of these points.
Using the midpoint formula from the formula booklet:
\(M=\left(\frac{1+4}{2},\ \frac{1+10}{2}\right)=\left(2.5,5.5\right)\)
A perpendicular bisector crosses a line segment CD. Given that point C = (-3, 8) and D = (-1, 2), find the gradient of the perpendicular bisector.
First find the gradient of the line segment CD.
Gradient of CD =\(\frac{8-2}{-3--1}=\frac{6}{-2}=-3\)
Hence, the gradient of the perpendicular bisector = \(\frac{-1}{-3}=\frac{1}{3}\)
Using the formula given in the "Prior learning -SL and HL" section of the formula booklet:
\(M=\left(\frac{3+7}{2},\ \frac{5+13}{2}\right)=\left(5,\ 9\right)\)