If A(4, –6), B(3, –2) and C(5, 2) are the vertices of ∆ABC, then verify the fact that a median of a triangle ABC divides it into two triangle of equal areas. - Mathematics

Advertisements
Advertisements
Sum

If A(4, –6), B(3, –2) and C(5, 2) are the vertices of ∆ABC, then verify the fact that a median of a triangle ABC divides it into two triangle of equal areas.

Advertisements

Solution

Let D be the mid-point of BC. Then, the coordinates of D are (4, 0).

`∴ "Area os " triangleABC = 1/2|(4xx(-2)+3xx2+5xx(-6))-(3xx(-6)+5xx(-2)+4xx2)|`

`⇒ "Area of "triangleABC=1/2|(-8+6-30)-(-18-10+8)|`

`⇒ "Area of "triangleABC =1/2|-32+20|=6" sq. units"`

Also, We have

`therefore" Also of "triangle ABD =|{(4xx(-2)+3xx0+4xx(-6))}-{3xx(-6)+4xx(-2)+4xx0}|`

`⇒ "Area of "triangle ABD = 1/2|(-8+0+26)-(-18-8+0)|`

`⇒"Area of "triangle ABD =1/2|(-32+26)|=3" sq. units"`

`\Rightarrow \frac{Area\ of\ \Delta ABC}{Area\ of\ \Delta ABD}=\frac{6}{3}=\frac{2}{1}`

⇒ Area of ∆ABC = 2 (Area of ∆ABD)

  Is there an error in this question or solution?

RELATED QUESTIONS

Find the values of k so that the area of the triangle with vertices (1, -1), (-4, 2k) and (-k, -5) is 24 sq. units.


Prove that the area of a triangle with vertices (t, t −2), (t + 2, t + 2) and (t + 3, t) is independent of t.


Find the area of the triangle PQR with Q(3,2) and the mid-points of the sides through Q being (2,−1) and (1,2).


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

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


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 (3, 1) and (9, 3) using determinant.


Find the area of the following triangle:


Find the missing value:

Base Height Area of triangle
15 cm ______ 87 cm2

Find the area of the quadrilaterals, the coordinates of whose vertices are

(1, 2), (6, 2), (5, 3) and (3, 4)


Find the centroid of the triangle whosw vertices is  (1,4), (-1,1) and (3,2) . 


Find the angle subtended at the origin by the line segment whose end points are (0, 100) and (10, 0).


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 ?


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


The perimeter of a triangular field is 540 m and its sides are in the ratio 25 : 17 : 12. Find the area of the triangle ?


Show that the points are the vertices of an isosceles right triangle.


Find the centroid of  ΔABC  whose vertices are A(-1, 0) B(5, -2) and C(8,2) 

 


Find the area of  ΔABC whose vertices are:
A(-5,7) , B (-4,-5) and C (4,5)

 


Find the area of  ΔABC with A(1, -4) and midpoints of sides through A being (2, -1) and (0, -1).


 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?


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


If the points (3, -2), (x, 2), (8, 8) are collinear, then find the value of x.


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 a triangle with base 4 cm and height 6 cm is 24 cm2.


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


The base and the corresponding altitude of a parallelogram are 10 cm and 3.5 cm, respectively. The area of the parallelogram is 30 cm2.


Find the cost of laying grass in a triangular field of sides 50 m, 65 m and 65 m at the rate of Rs 7 per m2.


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.


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.


Area of triangle MNO in the figure is ______.


In the given figure, if PR = 12 cm, QR = 6 cm and PL = 8 cm, then QM is ______.


In the given figure, ΔMNO is a right-angled triangle. Its legs are 6 cm and 8 cm long. Length of perpendicular NP on the side MO is ______.


Area of a right-angled triangle is 30 cm2. If its smallest side is 5 cm, then its hypotenuse 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.


In the given figure, ratio of the area of triangle ABC to the area of triangle ACD is the same as the ratio of base BC of triangle ABC to the base CD of ΔACD.


Ratio of the area of ∆WXY to the area of ∆WZY is 3:4 in the given figure. If the area of ∆WXZ is 56 cm2 and WY = 8 cm, find the lengths of XY and YZ.


In the given figure, triangle AEC is right-angled at E, B is a point on EC, BD is the altitude of triangle ABC, AC = 25 cm, BC = 7 cm and AE = 15 cm. Find the area of triangle ABC and the length of DB.


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 ______.


Using determinants, find the area of ΔPQR with vertices P(3, 1), Q(9, 3) and R(5, 7). Also, find the equation of line PQ using determinants.


Share
Notifications



      Forgot password?
Use app×