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You will learn math problems solutions for geometry as follows:

  • – the problems 23 - 27 are examples with solutions on the topic "Formula for calculating vector components from coordinates of the initial and terminal points";
  • – the problems 28 - 33 have answers for questions how to find coordinates of the midpoint of a segment, the vector magnitude from x and y components;
  • – solutions of the problems on the Cartesian coordinate plane are explained in the math tests 34 - 38;
  • – the topic "Points, triangles, quadrilaterals on the x-y coordinate plane" is presented in the problems 39 - 42 of the textbook;
  • – in the examples of problems 43-47 of the workbook, it is shown how to find various solutions for the topic "Equation of a circle";
  • – the question of how to calculate and write the equation of a line is considered in the tests 48 - 53.

The simplest problems on the x-y coordinate plane

Definition:

The position vector of an arbitrary point M is a vector connects the origin of the coordinate plane and the point M.

 

Theorem:

The coordinates of any point on the coordinate plane are equal to the corresponding components of the position vector of this point.

Given:

M(x;y) is any point

is the position vector of the point M

Prove: {x; y}

Proof:

Case 1 (Figure 1). If x>0

By the parallelogram rule , where х=OM1

= OM1 = x

= OM2 = y

So,

Therefore {x; y}

 

Case 2 (Figure 2). If x<0

 

 

 

 

Formula for calculating the vector components from coordinates of the initial and terminal points of this vector.

Theorem:

Each vector component is equal to the difference of the corresponding coordinates of the initial and terminal points of this vector.

Given:

is an arbitrary vector

A (a1;a2) are the coordinates of the initial point of the vector AB

B (b1;b2) are the coordinates of the terminal point of the vector AB

 

Prove: {b1 – a1; b2 – a2}

Proof:

By the triangle rule, the vector ==

= {b1 – a1; b2 – a2}, where {b1;b2} and {a1;a2}, since they are the position vectors.

***

 

Problem 23.

Given:

ΔABC is an isosceles triangle

AB=2a, CO=h

Find: the coordinates of the point A, the point B and the point C

Solution:

The point C lies on the ordinate axis, i.e., y-axis, we see that the coordinates of the point C(0;h)

Since CO is the height in the given isosceles triangle, we see that CO is the median too.

Therefore, AO=OB=2a : 2 = a

We see that the coordinates of the point A(–a;0), the coordinates of the point B(a;0), since A and B lie on the x-axis.

***

 

Problem 24.

Find: components of a vector given the coordinates of endpoints of the vector

1) A(2;7) and B(–2;7); 2) A(–5;1) and B(–5;27)

Solution:

1) = B–A={–2;7}–{2;7} = {–2–2;7–7}={–4;0}

2) = {–5;27}– {–5;1} = {–5+5;27–1}={0;26}

***

 

Problem 25.

Find: components of a vector given the coordinates of initial and terminal points of the vector

1) A(–3;0) and B(0;4); 2) A(0;3) and B(–4;0)

Solution:

1) = B–A=

2) = B–A=

***

 

Problem 26.

Find: components of a vector , and the coordinates of initial and terminal points of the vector

Solution:

1) If A(0;0), B(1;1), then = B–A=

2) If A(x;–3), B(2;–7), {5;y}, then = B – A =

5 = 2 – x x= –3

y = –4

3) If A(a;b), {c;d}, we have = B – A

B=

4) If A(1;2), {0;0}, we see that = B – A

B=

***

Problem 27.

Given:

MNPQ is a square

P (–3;3) are the coordinates of the point

NQ ∩ PM = O(0;0)

Find: the coordinates of points M,N,Q

Solution:

Since the diagonals in the square are divided at the intersection point in half, we see that PO=OM , since PO and OM are collinear vectors, but they are ↑↓ multidirectional vectors.

Then {3;–3} M(3;–3)

Draw a perpendicular joining the point P in the direction to the y-axis to the intersection with the diagonal NQ.

Then the point N is the vertex of the square. Since the vertex of the square is located in the first (I) coordinate quarter, we see that N(3;3);

Since the diagonals at the intersection point are divided in half, we see that NO=OQ

, since these vectors are collinear, but differently directed ↑↓ , i.e., they are collinear and oppositely directed vectors

Then {–3;–3} Q(–3;–3)

Answer: M(3;–3), Q(–3;–3), N(3;3) and

M(3;–3), N(–3;–3), Q(3;3)

***

 

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Formula how to find the coordinates of the midpoint of a segment

Each coordinate of the midpoint of a segment is equal to the half-sum of the corresponding coordinates of the initial and terminal points of the segment.

 

Problem 28.

Given:

The segment AB, A(x1; y1), B(x2; y2)

CAB, AC=CB

 

Prove: C

Proof:

Since the vector OC is equal to half the sum of two vectors OA and OB constructed at the same point O, we see that

Since and are the position vectors of points A and B, we see that {x1; y1} and {x2; y2}.

Then

C, since is the radius-vector of the point C.

Formula how to find the coordinates of the middle of a segment

C

 

Example.

Given:

The points A and B are endpoints of the segment AB, the point M is the midpoint of the segment AB

a) Given coordinates of points A(2; –3); B(–3; 1) how to find the coordinates of the midpoint of the segment AB

Using the formula to find the coordinate of the midpoint of the segment, we get M MM

b) Given coordinates of points B(4; 7); M(–3; –2), then how to find the coordinates of the point A(x;y).

and

–6 = 4+x and –4 = 7+y

x= –10 and y= –11

Answer: M; A (– 10; – 11)

***

 

Problem 29. How to find the vector magnitude or length from x and y components

Given: vector components

{a1; a2}

 

Prove that the magnitude or length of a vector is equal to the square root of sum of squares of x and y components

Proof:

Let us draw a vector at the origin O (0;0) =

But is a position vector of the point A. Then A(a1; a2)

Having used the triangle ΔOAA1 by the Pythagorean theorem:

a1=OA1; a2=OA2=AA1

OA2=

OA=

Formula how to find the magnitude of a vector from x and y components

 

Example.

{11;–11}, then =

=

{10;0}, then = 10

 

Problem 30.

Given: the coordinates of the point

M1(x1; y1), M2(x2; y2)

 

Find: distance between two points

d= M1M2 = ?

Solution:

{ x2– x1; y2– y1},

, = M1M2 = d

Then

Formula how to find the distance between two points

***

 

Problem 31.

 

Given: coordinates of segment points: A is the initial point, B is the terminal point, M is the midpoint of a segment

 

1

2

3

4

A

(2;–3)

?

(0;0)

(0;1)

B

(–3;1)

(4;7)

(–3;7)

?

M

?

(–3;–2)

?

(3;–5)

 

Find: coordinates of endpoints and the midpoint of the segment AB.

Solution:

Using the formula to find the coordinate of a midpoint of a segment, we obtain M

1) MM(–0,5 ; –1)

2) If B (4;7) and M (–3;–2), then we must find the coordinates of the initial point of the segment, i.e., A(x;y) – ?

–3=

–6=4+x

x=–10

 

–2=

–2•2=7+y

y=– 11

3) MM(–1,5 ; 3,5)

4) If A (0;1) and M (3;–5), then we must find the coordinates of the terminal of the segment, i.e., B(x;y) – ?

3=

6=x+0

x=6

 

–5=

2•(–5)=y+1

y=– 11

 

 

5

6

7

8

A

(c;d)

(3;5)

(1;3)

(3t+5;7)

B

?

(3;8)

?

(t+7;–7)

M

(a;b)

?

(0;0)

?

 

5) If A (c;d) and M (a;b), then we must find the coordinates of the terminal point of the segment, i.e., B(x;y) – ?

a=

2a=c+x

x=2a–c

 

b=

2b=d+y

y=2b–d

6) MM(3 ; 6,5)

7) If A (1;3) and M (0;0), then we must find the coordinates of the terminal point of the segment, i.e., B(x;y) – ?

0=

0=1+x

x=–1

 

0=

0=3+y

y=–3

8) MM(2t+6 ; 0)

***

 

Problem 32.

Given: vector components

{5;9}; {–3;4}; {–10;–10}; {10;17}

 

Find: the magnitude of vectors ; ; ; – ?

Solution:

Using the formula to calculate the magnitude of a vector from x and y components,

a) ====

b) =====5

c) ====10

d) == =

***

 

Problem 33.

Given: the coordinates of points A and B

 

1

2

3

4

A

(2;7)

(–5;1)

(–3;0)

(0;3)

B

(–2;7)

(–5;–7)

(0;4)

(–4;0)

d

?

?

?

?

 

Find: the distance between two points – ?

Solution:

Using the formula to calculate the distance between two points

, we obtain

1) d = = = 4

2) d = = = 8

3) d = ===5

4) d = ===5

***

 

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Solving problems on two-dimensional Cartesian plane

Problem 34.

Given:

a two-dimensional coordinate grid,

a triangle ΔABC,

AM is the median

coordinates of the vertices of a triangle

A(0;1), B(1;–4), C(5;2)

Find: the length of the median AM– ?

Solution:

Using the formula to find the coordinate of the midpoint of the segment, we get M MM(3 ; –1), since the point M is the midpoint of BC

Using the formula to calculate the distance between two points

, we obtain

AM ===

Answer: AM =

***

 

Problem 35.

Given:

OACB is a parallelogram

The length of side OA=a

The coordinates of the point B (b;c)

 

Find: the coordinates of the point C(x;y),

the length of sides AC, CO – ?

Solution:

Since the point O is the origin of the x-y coordinate plane, we see that the coordinates of the point O (0;0).

Since OA=a and the point A lies on the abscissa axis, i.e., x-axis, we see that the coordinates of the point A (a;0).

By the rule of the parallelogram and since OC is the vector, we see that

=

So, the vector has components {a+b;c}

Since is the position vector of the point C, we see that the coordinates of the point C (a+b;c).

Using the formula to calculate the distance between two points

, we obtain

AC ==

CO = ==

Answer: C (a+b;c) ; AC=; CO =

***

 

Problem 36.

Given:

a two-dimensional coordinate plane,

a triangle ,

the coordinates of the vertices of the triangle A(0;1), B(1;–4), C(5;2)

 

1) Prove: triangle ΔABC is isosceles

Proof:

Using the formula to calculate the distance between two points

, we obtain

AC===

AB===

Since AC=AB=, we see that triangle ΔABC is isosceles.

 

2) Find: SΔABC – the area of the triangle ΔABC – ?

Solution:

We draw the height AM to the base BC.

Using the formula for calculating the area of a triangle, we obtain

SΔABC= AM•BC

Using the formula to find the coordinate of the midpoint of the segment, we get M MM(3 ; –1), since the point M is the midpoint of BC

Using the formula to calculate the distance between two points

, we obtain

AM ===

BC ===2

Then the area of the triangle SΔABC =•2==13

Answer: SΔABC =13.

***

 

Problem 37.

Given:

the Cartesian coordinate plane,

a triangle ΔMNP,

the coordinates of vertices of a triangle

M(4;0), N(12;–2), P(5;–9)

 

Find: the perimeter of the triangle PΔMNP – ?

Solution:

Using the formula to calculate the distance between two points

, we obtain

MN====

NP===

PM===

Using the formula to calculate the perimeter of a triangle, we get PΔMNP =MN+NP+PM=++

Answer: PΔMNP =++

***

 

Problem 38.

Given:

MNPQ is a quadrilateral

coordinates of vertices or angles of a quadrilateral are as follows

M(1;1), N(6;1), P(7;4), Q(2;4)

 

1) Prove: MNPQ is a parallelogram

Proof:

Using the formula to calculate the distance between two points

, we obtain

MN=== 5

PQ === 5

NP ===

QM ===

Since MN=PQ=5 and NP=QM=, we see that MNPQ is a parallelogram

 

2) Find: the length of MP, NQ – ?

Solution:

Using the formula to calculate the distance between two points

, we obtain

MP === ==3

NQ === = 5

Answer: MP= 3, NQ = 5

***

 

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Points, triangles, quadrilaterals on the coordinate plane

Problem 39.

Given:

the coordinates of the points C(4;–3), D(8;1)

the point A lies on the y-axis

AC=AD

 

Find: the coordinates of the point A – ?

Solution:

1) Since the point A lies on the y-axis, i.e., the ordinate axis, we see that its coordinates are (0;y).

2) Using the formula to calculate the distance between two points , we obtain

AD==

3) AC==

4) Since AC=AD, we see that =

64 + (y – 1)2= 16 + (3 + y)2

64 + 1 – 2y+y2=16 + 9 + 6y + y2

–2y – 6y= 25 – 65

–8y = –40

y = 5

Answer: A (0; 5).

***

 

Problem 40.

Given:

O(0;0)

ΔABC is isosceles

the median OC=160cm

the base AB=80cm

 

Find: AK; NB

Solution:

1) Since the points A and B lie on the x-axis, i.e., the abscissa axis,

we see that AO=OB=80 : 2 = 40 (cm)

The coordinates of the points B(40;0) and A(–40;0)

2) Since OC = 160 cm and the point C lies on the y-axis

The coordinates of the point C(0;160)

3) Since K is the midpoint of BC, we see that, using the formula to find the coordinate of the midpoint of the segment, we obtain K

KK(20 ; 80)

4) Then, using the formula to calculate the distance between two points , we obtain

AK= = ==100 (cm)

5) Since N is the midpoint of the segment AC, we see that, using the formula to find the coordinate of the midpoint of the segment, we obtain

N NN(–20 ; 80)

6) Using the formula to calculate the distance between two points , we obtain

NB = = ==100 (cm)

Answer: AK = NB = 100 cm

***

 

Problem 41.

Given:

a triangle ΔABC

the right angle C = 90°

The point M lies on the side AB

The point M is the midpoint of the side AB

BC=a, AC=b

 

Prove: the length of the height in a right angled triangle is half the length of the hypotenuse.

Proof:

Since BC=a, AC=b, we see that

the point C (0;0) – since the point C is the origin of the coordinate plane

we have B(a;0), A(0;b).

Using the formula to find the coordinate of the middle of the segment, we obtain M MM(;), since the point M is the midpoint of AB.

Using the formula to calculate the distance between two points , we obtain

MC== =

AM==

Therefore, AM=MB=MC.

***

Problem 42.

Given:

ABCD is a parallelogram

AD=BC=a

the coordinates of points

B(b;c), D(a;0), C(a+b;c)

 

Prove:

The sum of squares of the sides of the parallelogram is equal to the sum of squares of diagonals of a parallelogram (where diagonals are the segments connecting the opposite vertices of the parallelogram).

AB2 + BC2 + CD2 + AD2 = AC2 + BD2

Proof:

Using the formula to calculate the distance between two points , we obtain

AB= AB2 = b2+c2

AD=AD2 = a2

AC=AC2 = (a+b)2 + c2

BD=AC2 = (a–b)2 + c2

Since AB=CD, AD=BC, we see that

AB2 + AD2 + BC2 + CD2 = 2(AB2 + AD2) = 2(b2+c2 + a2)

AC2 + BD2 = (a+b)2 + c2 + (a–b)2 + c2 = a2 + 2ab + b2 + c2 + a2 – 2ab + b2 + c2 = 2(a2 + b2 + c2)

We see that AB2 + BC2 + CD2 + AD2 = AC2 + BD2

 

***

 

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The equation of a line and the equation of a circle

Definition: An equation with two variables x and y is called an equation of the line L if the coordinates of any point belonging to the line L satisfy this equation, and the coordinates of any point not lying on this line does not satisfy this equation.

Theorem - the formula of the equation of a circle or the equation of the circle radius:

On a rectangular or Cartesian coordinate system, the equation of the circle of center at the point C(x0; y0) and the radius r has the form

(x – x0)2 + (y – y0)2 = r 2

 

Given:

The circle (C; r) of the center at the point C,

the coordinates of the point C(x0; y0)

 

Prove: the equation of the circle (x – x0)2 + (y – y0)2 = r 2

Proof:

Consider a point M(x;y), lying on the circle (C;r)

Using the formula to calculate the distance between two points , we obtain

CM=, but CM=r

Then we get CM2 = r2

r2 = (x – x0)2 + (y – y0)2 (1)

The coordinates of the point M satisfy equation (1).

If N(x1;y1) does not lie on the circle (C;r), then

Since NC ≠ r , we see that the coordinates of the point N do not satisfy equation (1).

Hence, the equation of the circle (x – x0)2 + (y – y0)2 = r 2

If the center of the circle, i.e. the point C has the coordinates C(x0; y0) = O(0;0) the equation of the circle is x2 + y2 = r 2

***

 

Problem 43.

Given: the equation of a circle with the center at the point A passing through 2 points

a) x2 + y2 = 9

b) (x – 1)2 + (y + 2)2 = 4

c) (x + 5)2 + (y – 3)2 = 25

d) (x – 1)2 + y2 = 4

e) x2 + (y + 2)2 = 2

 

Find: the center of the circle and its radius r

Solution:

a) Let there be a circle of center A and radius r or:

circle (A;r),

and a point B with coordinates (x;y) that lies on a given circle or:

B(x;y) circle.

 

But AB = r, therefore, AB2=r2 r2=9 r2=32, i.e. r = 3.

Since x2 + y2 = 9, we see that x0 =0 and y0=0, and we see that the center of the circle has the coordinates A(0;0). The graph of the circle is shown in Figure a).

 

 

b) Since (x – 1)2 + (y + 2)2 = 4, we see that by the equation of the circle

x0 = 1 and y0= – 2.

Then A(1;–2), where A is the center of the circle.

Since r2 = 4, we see that r = 2. The graph of the circle is shown in Figure b)

 

c) Since (x + 5)2 + (y – 3)2 = 25, we see that by the equation of the circle

x0 = –5 and y0= 3.

Then A(–5;3), where A is the center of the circle.

Since r2 = 25, we see that r = 5.

Draw this circle. The graph of the circle is shown in Figure c)

 

d) Since (x – 1)2 + y2 = 4, then by the equation of the circle

x0 = 1 and y0= 0.

Then A(1;0), where A is the center of the circle.

Since r2 = 4, we see that r = 2.

Draw this circle. The graph of the circle is shown in Figure d)

 

 

e) Since x2 + (y + 2)2 = 2, we see that by the equation of the circle

x0 = 0 and y0= –2.

Then A(0;–2), where A is the center of the circle.

Since r2 = 2, we see that r ≈ 1,4.

Draw this circle. The graph of the circle is shown in Figure e)

 

***

 

Problem 44.

Given: the circle is given by equation

a) x2 + y2 = 25

b) (x – 1)2 + (y + 3)2 = 9

 

Determine: which of the points A, B, C, D, E belong to the circle (A; r),

if given are the coordinates of the points A(3;–4); B(1;0); C(0;5); D(0;0); E(0;1)

Solution:

a) If A(3;–4), where x=3 and y=–4, we see that 32 + (–4)2 = 25

9+16=25

25=25

The point A belongs to the circle (A; r) or the point A circle (A;r)

 

If B(1;0), where x=3 and y=0, then 12 + 02 = 25

1≠25

Therefore, the point B does not belong to the circle (A; r) or the point B circle (A;r)

 

If C(0;5), where x=0 and y=5, then 02 + 52 = 25

25=25

This means that the point C belongs to the circle (A; r) or the point Ccircle (A;r)

 

If D(0;0), where x=0 and y=0, then 02 + 02 = 25

0≠25

This means that the point D does not belong to the circle (A; r) or the point Dcircle (A;r)

 

If E(0;1), where x=0 and y=1, then 02 + 12 = 25

1≠25

Hence, the point E does not belong to the circle (A; r) or the point Ecircle (A;r)

 

b) If A(3;–4), where x=3 and y=–4, then (3 – 1)2 + (–4 + 3)2 = 9

4+1=9

5≠9

The point A does not belong to a circle (A; r) or the point Acircle (A;r)

 

If B(1;0), where x=1 and y=0, then (1 – 1)2 + (0 + 3)2 = 9

9=9

The point B belongs to the circle or the point Bcircle (A;r)

 

If C(0;5), where x=0 and y=5, then (0 – 1)2 + (5 + 3)2 = 9

1+64≠9

65≠9

The point C does not belong to the circle (A; r) or the point Ccircle (A;r)

 

If D(0;0), where x=0 and y=0, then (0 – 1)2 + (0 + 3)2 = 9

10≠9

The point D does not belong to the circle (A; r) or the point Dcircle (A;r)

 

If E(0;1), where x=0 and y=1, then (0 – 1)2 + (1 + 3)2 = 9

17≠9

The point E does not belong to a circle (A; r) or the point Ecircle (A;r)

***

 

Problem 45.

Given: the circle given by equation

The equation of a circle through 2 points (x+5)2 + (y – 1)2 = 16

The circle (C;r), where r=4, C(–5;1)

The points A(–2;4), B(–5;–3)

 

Determine: which of the points A or B belong to the circle (C;r)

Solution:

Using the formula to calculate the distance between two points , we obtain

CA===3,

3 >4, then the point A is outside the circle (C;r)

 

CB===4,

4=4, then the point B lies on the circle (C;r)

***

 

Problem 46.

Given:

the circle (C;r), where the diameter of the circle d=MN

the coordinates of the points M(–3;5), N(7;–3)

 

Write: the equation of a circle of center C passing through 2 points M and N

Solution:

Using the formula to calculate the distance between two points , we obtain

CA====2,

Since d=2r, then r =• d = =

Using the formula to find the coordinate of the middle of the segment, we obtain the coordinates of the center of the circle C CC(2;1), since the point C is the middle of MN.

 

Using the equation of the circle equation (x – x0)2 + (y – y0)2 = r 2, therefore, (x – 2)2 + (y – 1)2 =

(x – 2)2 + (y – 1)2 = 41

Answer: (x – 2)2 + (y – 1)2 = 41

***

 

Problem 47.

Given:

the circle (C;r), where the coordinates of the center of the circle C(0;y)

The points A and B lie on a circle

The coordinates of the points A(–3;0), B(0;9)

 

Write: the equation of a circle of center at the point C passing through 2 points

Solution:

Using the equation of the circle equation (x – x0)2 + (y – y0)2 = r 2, therefore, x2 + (y – y0)2 = r 2

Since the point B lies on the circle and its coordinates B(x=0;y=9).

Then the point B lies on the ordinate axis, i.e., the y-axis, then CB=r.

x2 + (y – y0)2 = r 2 02+ (9 – y0) 2 = r 2

 

Since the point A lies on the circle and its coordinates

A(x=–3;y=0).

Then the point A lies on the abscissa axis, i.e., the x-axis, then CB=r.

(–3)2 + (0 – y0)2 = r 2 9 + y02 = r 2

 

Then (9 – y0) 2 = 9 + y02

y0 2 – 18 y0 + 81 = 9 + y02

– 18 y0 = – 72

y0 = 4

Therefore, C(0;4).

We obtain the equation of a circle of the form: x2 + (y – 4)2 = r 2

Let us find the radius r.

Since the points A and B belong to the circle, then

(–3)2 + (0 – 4)2 = r2

9+16 = r2

r2 = 25

or

02 + (9 – 4)2 = r2

r2 = 25

 

So we get the equation of the circle x2 + (y – 4)2 = 25

Answer: x2 + (y – 4)2 = 25

***

 

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The equation of the straight line

The graph of the equation of a straight line passing through a point. Types of equations of the line.

1) the equation of the straight line l: ax + by + c = 0, where a,b,c are the coefficients of the equation of the straight line
2) l1: y = y0 is the equation of the straight line passing through the point M0, perpendicular to the straight line y, the y-axis, the ordinate axis, parallel to the straight line x, the x-axis, the abscissa axis
3) l2: x = x0 is the equation of a straight line passing through the point M0, parallel to the straight axis of ordinates Oy, perpendicular to the x-axis, i.e. the axis of abscissas
4) y = 0 the equation of the straight line, the x-axis, i.e., the axis of abscissas, passing through the origin point of the coordinate plane
5) x = 0 the equation of the y-axis, i.e. the ordinate axis that passes through the origin point of the coordinate plane

 

 

Problem 48.

Write: the equation of a straight line passing through one point, i.e. the center of a circle.

a) (x +3)2 + (y – 2)2 = 25, parallel to the y-axis

b) (x –2)2 + (y +5)2 = 3, parallel to the x-axis

Solution:

a) (x +3)2 + (y – 2)2 = 25 the center of the circle (-3; 2), where r=5

The straight line passes parallel to the y-axis, then its equation is x=x0.

Therefore, x= –3.

 

b) (x – 2)2 + (y +5)2 = 3 the center of the circle (2;–5), where r=

The straight line passes parallel to the x-axis, then its equation is y=y0.

Therefore, y= –5.

***

 

Problem 49.

Given: the equations of two straight lines

b1: 4x + 3y – 6=0

the straight line b2 is given by the equation: 2x + y – 4=0

 

Find: the coordinates A(x;y), where A is the intersection point of the straight lines b1 and b2

Solution:

We will compose and solve the system of equations of the straight lines

The sum of equations (1) and (2) –2x = –6

Then x=3.

We substitute x=3 in the equation 4x+3y=6.

Then 12+3y= –6 y= –2

Answer: A(3; –2)

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Problem 50.

Given: the coordinates of two points A(1;–1), B(–3;2)

Find the equation of the straight line AB having used the given points

Solution:

The general equation of the straight line has the form AB=ax+by+c=0, where having used the coordinates, it is necessary to find a=? b=? c=?

Since the points A and B lie on the straight line AB (or: A and B AB),

then their coordinates satisfy the equation of the straight line AB

 

–a=–3c a=3c

Then we substitute a=3c in the equation a–b=–c. Therefore,

3c–b=–c

–b=–4c

b=4c

Then the equation of the line AB through two points: ax+by+c=0

3c+4cy+c=0

c(3x+4y+1)=0

3x+4y+1=0

Answer: the equation of a straight line having used two given points is 3x+4y+1=0

***

Problem 51.

Given: the coordinates of two points C(2;5), D(5;2)

Solve: find the equation of the line AB passing through 2 points

Solution:

The general equation of the line AB: ax+by+c=0, where it is necessary to find a=? b=? c=?

Since the points A and B lie on the line AB, then their coordinates satisfy the equation of the line AB

 

–10,5b=1,5c

c=–7b

Then in the equation 2a+5b=–c we substitute

c=–7b, then, 2a+5b=7b

2a=7b–5b

a=b

Then the equation of the line AB should be written: ax+by+c=0

bx+by+(–7b)=0

b(x + y – 7)=0

x + y – 7=0

Answer: for these two points, the equation of the line AB is x+y–7=0

***

Problem 52.

Given:

the line AB is given by the equation 3x–4y+12=0

Find: the coordinates of two points A and B that are the points of intersection with the coordinate axes

Solution:

Since the points A and B lie on the line AB, therefore their coordinates satisfy the equation of the straight line AB.

Since the line AB intersects with the coordinate axes, then

coordinate I – (x;0)

coordinate II – (0;y)

Therefore, 3x–4•0+12=0

3x=–12

x=–4 A(–4;0)

 

3•0–4y+12=0

–4y=–12

y=3 B(0;3)

Next, we construct a straight line AB in the coordinate plane.

To construct a straight line, we first mark the points in the coordinate system: the abscissa of the point A is -4, its ordinate is zero; the abscissa of the point B is zero, its ordinate is 3. Draw a straight line through the marked two points.

Answer: A(–4;0), B(0;3)

***

 

Problem 53.

Given:

Triangle ΔABC

The coordinates of the vertices of the triangle A(4;6), B(–4;0), C(–1;–4)

The point M lies on the segment AB

Solve: find the equation of a straight line containing the median MC, having used the coordinates of the points

Solution:

Let us find the coordinate of the point M.

Using the formula to find the coordinate of the middle of the segment, we obtain M M

 

M(0;3), since the point M is the midpoint of AB.

The canonical equation of the straight line MC passing through the point has the form: ax+by+c=0, where it is necessary to find a=? b=? c=?

Since the points M and C lie on the line MC (or: M,CMC), then their coordinates satisfy the equation of the straight line MC.

–3a=–7c

Having substituted

in the equation –a – 4b = –c, we obtain

b=

Knowing a and b, we find the coordinate equation of the straight line MC: ax+by+c=0

7x–y+3=0

Answer: the linear equation of the straight line MC: 7x–y+3=0

***