Advertisements
Advertisements
प्रश्न
Write the area of the triangle with vertices at (a, b + c), (b, c + a) and (c, a + b).
Advertisements
उत्तर
Let A(a, b + c), B(b, c + a) and C(c, a + b) be the vertices of the the given triangle.
\[\therefore\text{ Area of ∆ ABC }= \frac{1}{2}\left| x_1 \left( y_2 - y_3 \right) + x_2 \left( y_3 - y_1 \right) + x_3 \left( y_1 - y_2 \right) \right|\]
\[ = \frac{1}{2}\left| a\left( c + a - a - b \right) + b\left( a + b - b - c \right) + c\left( b + c - c - a \right) \right|\]
\[ = \frac{1}{2}\left| a\left( c - b \right) + b\left( a - c \right) + c\left( b - a \right) \right|\]
\[ = \frac{1}{2}\left| ac - ab + ab - bc + bc - ac \right|\]
\[ = 0\]
Hence, area of the triangle with vertices at (a, b + c), (b, c + a) and (c, a + b) is 0.
APPEARS IN
संबंधित प्रश्न
The vertices of a triangle ABC are A (0, 0), B (2, −1) and C (9, 2). Find cos B.
Four points A (6, 3), B (−3, 5), C (4, −2) and D (x, 3x) are given in such a way that \[\frac{\Delta DBC}{\Delta ABC} = \frac{1}{2}\]. Find x.
The points A (2, 0), B (9, 1), C (11, 6) and D (4, 4) are the vertices of a quadrilateral ABCD. Determine whether ABCD is a rhombus or not.
The base of an equilateral triangle with side 2a lies along the y-axis, such that the mid-point of the base is at the origin. Find the vertices of the triangle.
Find the equation of the locus of a point which moves such that the ratio of its distances from (2, 0) and (1, 3) is 5 : 4.
A (5, 3), B (3, −2) are two fixed points; find the equation to the locus of a point P which moves so that the area of the triangle PAB is 9 units.
If A (−1, 1) and B (2, 3) are two fixed points, find the locus of a point P, so that the area of ∆PAB = 8 sq. units.
A rod of length l slides between two perpendicular lines. Find the locus of the point on the rod which divides it in the ratio 1 : 2.
If O is the origin and Q is a variable point on y2 = x, find the locus of the mid-point of OQ.
What does the equation (x − a)2 + (y − b)2 = r2 become when the axes are transferred to parallel axes through the point (a − c, b)?
What does the equation (a − b) (x2 + y2) −2abx = 0 become if the origin is shifted to the point \[\left( \frac{ab}{a - b}, 0 \right)\] without rotation?
Find what the following equation become when the origin is shifted to the point (1, 1).
x2 + xy − 3x − y + 2 = 0
Find what the following equation become when the origin is shifted to the point (1, 1).
x2 − y2 − 2x +2y = 0
Find what the following equation become when the origin is shifted to the point (1, 1).
xy − x − y + 1 = 0
Find what the following equation become when the origin is shifted to the point (1, 1).
xy − y2 − x + y = 0
To what point should the origin be shifted so that the equation x2 + xy − 3x − y + 2 = 0 does not contain any first degree term and constant term?
Find what the following equation become when the origin is shifted to the point (1, 1).
x2 + xy − 3y2 − y + 2 = 0
Find what the following equation become when the origin is shifted to the point (1, 1).
xy − x − y + 1 = 0
Find what the following equation become when the origin is shifted to the point (1, 1).
x2 − y2 − 2x + 2y = 0
Verify that the area of the triangle with vertices (4, 6), (7, 10) and (1, −2) remains invariant under the translation of axes when the origin is shifted to the point (−2, 1).
The vertices of a triangle are O (0, 0), A (a, 0) and B (0, b). Write the coordinates of its circumcentre.
In Q.No. 1, write the distance between the circumcentre and orthocentre of ∆OAB.
Write the coordinates of the orthocentre of the triangle formed by points (8, 0), (4, 6) and (0, 0).
Three vertices of a parallelogram, taken in order, are (−1, −6), (2, −5) and (7, 2). Write the coordinates of its fourth vertex.
If the coordinates of the sides AB and AC of ∆ABC are (3, 5) and (−3, −3), respectively, then write the length of side BC.
Write the coordinates of the in-centre of the triangle with vertices at (0, 0), (5, 0) and (0, 12).
If the points (1, −1), (2, −1) and (4, −3) are the mid-points of the sides of a triangle, then write the coordinates of its centroid.
