Advertisements
Advertisements
प्रश्न
The vertices of a triangle ABC are A (0, 0), B (2, −1) and C (9, 2). Find cos B.
Advertisements
उत्तर
We know that
\[\cos B = \frac{a^2 + c^2 - b^2}{2ac}\], where a = BC, b = CA and c = AB are the lengths of the sides of ∆ ABC
Thus,
\[a = BC = \sqrt{\left( 2 - 9 \right)^2 + \left( - 1 - 2 \right)^2} = \sqrt{49 + 9} = \sqrt{58}\]
APPEARS IN
संबंधित प्रश्न
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.
Find the coordinates of the centre of the circle inscribed in a triangle whose vertices are (−36, 7), (20, 7) and (0, −8).
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 a point on the x-axis, which is equidistant from the points (7, 6) and (3, 4).
A point moves so that the difference of its distances from (ae, 0) and (−ae, 0) is 2a. Prove that the equation to its locus is \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\]
Find the locus of a point such that the sum of its distances from (0, 2) and (0, −2) is 6.
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.
Find the locus of a point such that the line segments with end points (2, 0) and (−2, 0) subtend a right angle at that point.
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.
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)?
Find what the following equation become when the origin is shifted to the point (1, 1).
x2 + xy − 3x − y + 2 = 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).
xy − y2 − x + y = 0
Find the point to which the origin should be shifted after a translation of axes so that the following equation will have no first degree terms: y2 + x2 − 4x − 8y + 3 = 0
Find the point to which the origin should be shifted after a translation of axes so that the following equation will have no first degree terms: x2 + y2 − 5x + 2y − 5 = 0
Find the point to which the origin should be shifted after a translation of axes so that the following equation will have no first degree terms: x2 − 12x + 4 = 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.
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 circumcentre of a triangle whose centroid and orthocentre are at (3, 3) and (−3, 5), respectively.
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.
Write the area of the triangle with vertices at (a, b + c), (b, c + a) and (c, a + b).
