The coordinates of A, B, C are (6, 3), (–3, 5) and (4, – 2) respectively and P is any point (x, y). Show that the ratio of the areas of triangle PBC and ABC is - Mathematics

Advertisements
Advertisements
Sum

The coordinates of A, B, C are (6, 3), (–3, 5) and (4, – 2) respectively and P is any point (x, y). Show that the ratio of the areas of triangle PBC and ABC is

Advertisements

Solution

We have,

`∴ "Area of ∆PBC" = \frac { 1 }{ 2 } |(5x+6+4y)–(–3y+20–2x)|`

`⇒ "Area of ∆PBC" = \frac { 1 }{ 2 } |5x + 6 + 4y + 3y – 20 + 2x|`

`⇒ "Area of ∆PBC" = \frac { 1 }{ 2 } |7x + 7y – 14|`

`⇒ "Area of ∆PBC" = \frac { 7 }{ 2 } |x + y– 2|`

`⇒ "Area of ∆PBC" = \frac { 7 }{ 2 } |6 + 3 – 2|`

`"[Replacing x by 6 and y = 3 in Area of Delta PBC]"`

`⇒ "Area of ∆ABC" = \frac { 49 }{ 2 }`

`\therefore \frac{Area\ of\ \Delta PBC}{Area\ of\ \Delta ABC}=\frac{\frac{7}{2}|x+y-2|}{\frac{49}{2}}`

`\therefore \frac{Area\ of\ \Delta PBC}{Area\ of\ \Delta ABC}=(x+y-2)/7`

  Is there an error in this question or solution?

RELATED QUESTIONS

If A(−4, 8), B(−3, −4), C(0, −5) and D(5, 6) are the vertices of a quadrilateral ABCD, find its area.


Find the area of a triangle whose vertices are A(3, 2), B (11, 8) and C(8, 12).


The vertices of ∆ABC = are A (4, 6), B(1, 5) and C(7, 2). A line is drawn to intersect sides AB and AC at D and E respectively such that `\frac{AD}{AB}=\frac{AE}{AC}=\frac{1}{4}` .Calculate the area of ∆ADE and compare it with the area of ∆ABC


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).


ABCD is a rectangle formed by the points A(-1, -1), B(-1, 4), C(5, 4) and D(5, -1). P, Q, R and S are the midpoints of AB, BC, CD and DA respectively. Is the quadrilateral PQRS a square? a rectangle? or a rhombus? Justify your answer.


Find the area of a triangle with vertices at the point given in the following:

(1, 0), (6, 0), (4, 3)


Find the area of a triangle with vertices at the point given in the following:

(2, 7), (1, 1), (10, 8)


Find the area of a triangle with vertices at the point given in the following:

(−2, −3), (3, 2), (−1, −8)


Show that points A (a, b + c), B (b, c + a), C (c, a + b) are collinear.


Find values of k if area of triangle is 4 square units and vertices are (k, 0), (4, 0), (0, 2)


Find values of k if area of triangle is 4 square units and vertices are (−2, 0), (0, 4), (0, k)


Find equation of line joining (1, 2) and (3, 6) using the determinant.


Find equation of line joining (3, 1) and (9, 3) using determinant.


If area of triangle is 35 square units with vertices (2, −6), (5, 4), and (k, 4), then k is ______.


Find the area of the following triangle:


ΔABC is right angled at A (see the given figure). AD is perpendicular to BC. If AB = 5 cm, BC = 13 cm and AC = 12 cm, Find the area of ΔABC. Also find the length of AD.


If A(–5, 7), B(–4, –5), C(–1, –6) and D(4, 5) are the vertices of a quadrilateral, find the area of the quadrilateral ABCD


The vertices of ΔABC are (−2, 1), (5, 4)  and (2, −3)  respectively. Find the area of the triangle and the length of the altitude through A.


If `a≠ b ≠ c`, prove that the points (a, a2), (bb2), (cc2) can never be collinear.


Find the area of a triangle whose sides are respectively 150 cm, 120 cm and 200 cm ?


Find the area of a triangle whose sides are 9 cm, 12 cm and 15 cm ?


Find the area of a triangle two sides of which are 18 cm and 10 cm and the perimeter is 42cm ?


In a ΔABC, AB = 15 cm, BC = 13 cm and AC = 14 cm. Find the area of ΔABC and hence its altitude on AC ?


Find the area of the blades of thc magnetic compass shown in Fig.. 12.27. (Take √11 = 3.32).


A(6,1) , B(8,2) and C(9,4) are the vertices of a parallelogram ABCD. If E is the midpoint of DC, find the area of ΔADE 


If the area of triangle ABC formed by A(x, y), B(1, 2) and C(2, 1) is 6 square units, then prove that x + y = 15 ?


 Using determinants, find the values of k, if the area of triangle with vertices (–2, 0), (0, 4) and (0, k) is 4 square units. 


In ☐ ABCD, l(AB) = 13 cm, l(DC) = 9 cm, l(AD) = 8 cm, find the area of ☐ ABCD.


Using integration, find the area of the triangle whose vertices are (2, 3), (3, 5) and (4, 4).


Using integration, find the area of triangle ABC, whose vertices are A(2, 5), B(4, 7) and C(6, 2).


What is the area of a triangle with base 4.8 cm and height 3.6 cm?


Find the area of the following triangle:


Find the area of the following triangle:


In ∆PQR, PR = 8 cm, QR = 4 cm and PL = 5 cm.

Find:
(i) the area of the ∆PQR
(ii) QM.


If the sides of a triangle are 3 cm, 4 cm and 5 cm, then the area is 


The table given below contains some measures of the right angled triangle. Find the unknown values.

Base Height Area
20 cm 40 cm ?

The table given below contains some measures of the right angled triangle. Find the unknown values.

Base Height Area
5 feet ? 20 sq.feet

The table given below contains some measures of the right angled triangle. Find the unknown values.

Base Height Area
? 12 m 24 sq.m

A field is in the shape of a right angled triangle whose base is 25 m and height 20 m. Find the cost of levelling the field at the rate of ₹ 45 per sq.m2


In a triangle ABC, if `|(1, 1, 1),(1 + sin"A", 1 + sin"B", 1 + sin"C"),(sin"A" + sin^2"A", sin"B" + sin^2"B", sin"C" + sin^2"C")|` = 0, then prove that ∆ABC is an isoceles triangle.


Let ∆ = `|("A"x, x^2, 1),("B"y, y^2, 1),("C"z, z^2, 1)|`and ∆1 = `|("A", "B", "C"),(x, y, z),(zy, zx, xy)|`, then ______.


If A, B, C are the angles of a triangle, then ∆ = `|(sin^2"A", cot"A", 1),(sin^2"B", cot"B", 1),(sin^2"C", cot"C", 1)|` = ______.


If the co-ordinates of the vertices of an equilateral triangle with sides of length ‘a’ are (x1, y1), (x2, y2), (x3, y3), then `|(x_1, y_1, 1),(x_2, y_2, 1),(x_3, y_3, 1)|^2 = (3"a"^4)/4`


Show that the points (a + 5, a – 4), (a – 2, a + 3) and (a, a) do not lie on a straight line for any value of a.


Show that the ∆ABC is an isosceles triangle if the determinant

Δ = `[(1, 1, 1),(1 + cos"A", 1 + cos"B", 1 + cos"C"),(cos^2"A" + cos"A", cos^2"B" + cos"B", cos^2"C" + cos"C")]` = 0


The area of a triangle with vertices (–3, 0), (3, 0) and (0, k) is 9 sq.units. The value of k will be ______.


The value of the determinant `abs((1,"x","x"^3),(1,"y","y"^3),(1,"z","z"^3))` is ____________.


If the points (a1, b1), (a2, b2) and(a1 + a2, b1 + b2) are collinear, then ____________.


If the points (2, -3), (k, -1), and (0, 4) are collinear, then find the value of 4k.


Find the area of the triangle whose vertices are (-2, 6), (3, -6), and (1, 5).


Let `Delta = abs (("x", "y", "z"),("x"^2, "y"^2, "z"^2),("x"^3, "y"^3, "z"^3)),` then the value of `Delta` is ____________.


The area of ∆ABC is 8 cm2 in which AB = AC = 4 cm and ∠A = 90º.


Find the area of the trapezium PQRS with height PQ given in figure


The area of a trapezium is 475 cm2 and the height is 19 cm. Find the lengths of its two parallel sides if one side is 4 cm greater than the other.


A rectangular plot is given for constructing a house, having a measurement of 40 m long and 15 m in the front. According to the laws, a minimum of 3 m, wide space should be left in the front and back each and 2 m wide space on each of other sides. Find the largest area where house can be constructed.


The dimensions of a rectangle ABCD are 51 cm × 25 cm. A trapezium PQCD with its parallel sides QC and PD in the ratio 9:8, is cut off from the rectangle as shown in the figure. If the area of the trapezium PQCD is `5/6` h part of the area of the rectangle, find the lengths QC and PD.


Find the missing value:

Base Height Area of parallelogram
______ 8.4 cm 48.72 cm2

Area of triangle MNO in the figure is ______.


If area of a triangular piece of cardboard is 90 cm2, then the length of altitude corresponding to 20 cm long base is ______ cm.


Triangles having the same base have equal area.


Area of a triangle PQR right-angled at Q is 60 cm2 in the figure. If the smallest side is 8 cm long, find the length of the other two sides.


Let a vector `αhati + βhatj` be obtained by rotating the vector `sqrt(3)hati + hatj` by an angle 45° about the origin in counter-clockwise direction in the first quadrant. Then the area of triangle having vertices (α, β), (0, β) and (0, 0) is equal to ______.


If (a, b), (c, d) and (e, f) are the vertices of ΔABC and Δ denotes the area of ΔABC, then `|(a, c, e),(b, d, f),(1, 1, 1)|^2` is equal to ______.


Share
Notifications



      Forgot password?
Use app×