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Equation of straight line
SLIDE 1
Revision
and are two points in the xy-plane. Find the length AB between the two points, their mid-point M, and the gradient, m, of the line joining them.
SLIDE 2
length,
mid-point,
gradient,
SLIDE 3
are two points. State in terms of their coordinates
? The length
? The mid-point, M, of the line joining A to B
? The gradient, m, of the line joining A to B
Modulus
The expression AB usually denotes the length of the line segment joining points A to B. An alternative symbol is , which makes it explicit that it is the length that is intended and not something else. The expression is read in full “the length of the line segment AB.” The symbol is also called the modulus of AB.
SLIDE 4
length,
mid-point,
gradient,
SLIDE 5
length,
mid-point,
gradient,
If you reverse the order of the subscripts in each of these formulas, so that, for example, we write , how, if at all, does that affect their values. Give reasons for your answers.
SLIDE 6
?
?
?
If the order of the subscripts is reversed, this does not change the value of any of these formulas, but the reason for this is different in each case. In ?, squares are always positive. In ?, the order in which you add numbers does not change the value. In ?, if you multiply the denominator and numerator of a fraction by , you leave the value of the fraction unchanged. In ?, if you reverse the order of the subscripts in the numerator, you must reverse them n the denominator. The order of the subscripts in the other two formulas does not matter.
SLIDE 7
Revision
? State the general form of the equation of the straight line.
? Find the equation of the straight line passing through the points .
SLIDE 8
?
?
SLIDE 9
Biconditional
The symbol is called the biconditional. If two statements P and Q are such that if P is true, then Q is true, and vice-verse, if Q is true, then P is true, then we write .
This also means .
In words, the biconditional is expressed by the phrase, if, and only if. This is abbreviated to iff.
It is customary, but not essential, to write iff in bold.
Yet, another way of expressing the same idea is by the word “equivalent”. means P is equivalent to Q.
P if, and only if Q
P is equivalent to Q
have (in this context) the same meaning. The term “equivalent” is used in several ways in mathematics and logic, and this is one of its uses only.
SLIDE 10
Two lines, , have equations
Using the biconditional
? State a condition that would show that are parallel.
? State a condition that would show that are perpendicular.
Write both statements using the expressions “if, and only if” and the symbol iff.
SLIDE 11
? parallel
parallel if, and only if,
parallel iff
? perpendicular
perpendicular if, and only if,
perpendicular iff
SLIDE 12
perpendicular
Give a graphical proof of the truth of this statement.
SLIDE 13
perpendicular
SLIDE 14
Find the equation of the perpendicular bisector of the line joining the points
SLIDE 15
Find the equation of the perpendicular bisector of the line joining the points
Solution
SLIDE 16
By finding the lengths of its sides, show that the quadrilateral formed by the points is a rhombus.
SLIDE 17
By finding the lengths of its sides, show that the quadrilateral formed by the points is a rhombus.
Solution
A rhombus is a quadrilateral where the sides are all equal. The student is also reminded of the shape of a kite.
We have to find the lengths of each side of the quadrilateral and check whether they are equal.
Therefore, the quadrilateral formed by the points A, B, C and D is a rhombus.
SLIDE 18
Find the point, M, that divides the line segment AB, where and , in the ratio . M is closer to B than A.
SLIDE 19
Find the point, M, that divides the line segment AB, where and , in the ratio . M is closer to B than A.
Solution
We divide both the distance of the x-coordinates, and the distance of the y-coordinates in the ratio . One way to look at this is that the point M is given by adding of the distance to the coordinates of A, but this is the same as a weighted average of the coordinates.
SLIDE 20
? Prove that if the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.
? By showing that its diagonals bisect each other, show that the quadrilateral formed by the points , , and is a parallelogram.
SLIDE 21
? Prove that if the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.
Solution
The proof is by congruent triangles. If we double up two triangles as shown in the diagram, then the two angles must be same, as well as the lengths of the third side, and we construct a parallelogram.
? By showing that its diagonals bisect each other, show that the quadrilateral formed by the points , , and is a parallelogram.
Solution
For two diagonals to bisect they must share the same midpoint.
Hence, ABCD is a parallelogram.
SLIDE 22
Show that the points form a right-angled triangle.
SLIDE 23
Show that the points form a right-angled triangle.
Solution
These points will form a right-angle triangle if the lengths obey Pythagoras’s Theorem, that the sum of the squares of the two shorter sides is equal to the square of the longer side.
Hence, ABC is a right-angled triangle.
Alternative. Show that the lines AB and BC are perpendicular using .
SLIDE 24
Determine whether AB and CD are parallel, perpendicular, or neither.
?
?
?
?
?
SLIDE 25
?
AB and CD are parallel
?
AB and CD are neither parallel nor perpendicular.
?
AB and CD are perpendicular
?
AB and CD are parallel
?
AB and CD are perpendicular
SLIDE 26
Given that the perpendicular bisector of the line joining A and B meets the y-axis at C, find the coordinates of C.
SLIDE 27
Given that the perpendicular bisector of the line joining A and B meets the y-axis at C, find the coordinates of C.
Solution
SLIDE 28
Find the equation of the line which goes through and is inclined at to the negative direction of the x-axis.
SLIDE 29
Find the equation of the line which goes through and is inclined at to the negative direction of the x-axis.
Solution
Find the equation of the line which goes through and is inclined at to the negative direction of the x-axis.
SLIDE 30
Find the equation of the perpendicular bisector of the line joining the points and .
SLIDE 31
Find the equation of the perpendicular bisector of the line joining the points and .
Solution
SLIDE 32
The equation of two sides of a square are and . If is one vertex of the square, find the coordinates of the other vertices.
SLIDE 33
The equation of two sides of a square are and . If is one vertex of the square, find the coordinates of the other vertices.
Solution
We then use similar triangles to find the other two points
SLIDE 34
The straight line passes through the points and .
? Find an equation for in the form
The equation has equation .
? Find the coordinates of the point of intersection of and .
SLIDE 35
The straight line passes through the points and .
? Find an equation for in the form
The equation has equation .
? Find the coordinates of the point of intersection of and .
Solution
?
?
SLIDE 36
The curve C has equation and the line L has equation . Show that C and L do not intersect.
SLIDE 37
The curve C has equation and the line L has equation . Show that C and L do not intersect.
Solution
Method. Attempt to solve simultaneously and . Expect a contradiction.
Square root of a negative number no real solutions
There, C and L do not intersect.
SLIDE 38
The diagram shows a rectangle ABCD whose diagonals meet at M. The coordinates of A and C are and respectively, and the equation of AB is . Find the equations of AD and CD.
SLIDE 39
SLIDE 40
The diagram shows a rectangle ABCD whose diagonals meet at M. The coordinates of A and C are and respectively, and the equation of AB is . The equations of AD and CD are . Find
? the coordinates of M, D and B
? the area of triangle BMC.
SLIDE 41
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