Advertisements
Advertisements
प्रश्न
If AD is the median of ∆ABC, using vectors, prove that \[{AB}^2 + {AC}^2 = 2\left( {AD}^2 + {CD}^2 \right)\]
Advertisements
उत्तर
Taking A as the origin, let the position vectors of B and C be \[\vec{b}\] and \[\vec{c}\] respectively.
It is given that AD is the median of ∆ABC.
∴ Position vector of mid-point of BC = \[\vec{AD} = \frac{\vec{b} + \vec{c}}{2}\]................(Mid-point formula)
Now,
\[{AB}^2 + {AC}^2 = \left| \vec{AB} \right|^2 + \left| \vec{AC} \right|^2 = \left| \vec{b} \right|^2 + \left| \vec{c} \right|^2\]
Also,
\[2\left( {AD}^2 + {CD}^2 \right)\]
\[ = 2\left( \left| \vec{AD} \right|^2 + \left| \vec{CD} \right|^2 \right)\]
\[ = 2\left[ \left( \frac{\vec{b} + \vec{c}}{2} \right) . \left( \frac{\vec{b} + \vec{c}}{2} \right) + \left( \frac{\vec{b} + \vec{c}}{2} - \vec{c} \right) . \left( \frac{\vec{b} + \vec{c}}{2} - \vec{c} \right) \right]\]
\[ = 2\left[ \left( \frac{\vec{b} + \vec{c}}{2} \right) . \left( \frac{\vec{b} + \vec{c}}{2} \right) + \left( \frac{\vec{b} - \vec{c}}{2} \right) . \left( \frac{\vec{b} - \vec{c}}{2} \right) \right]\]
\[ = \frac{\left| \vec{b} \right|^2 + 2 \vec{b} . \vec{c} + \left| \vec{c} \right|^2}{2} + \frac{\left| \vec{b} \right|^2 - 2 \vec{b} . \vec{c} + \left| \vec{c} \right|^2}{2}\]
\[ = \frac{2 \left| \vec{b} \right|^2 + 2 \left| \vec{c} \right|^2}{2}\]
\[ = \left| \vec{b} \right|^2 + \left| \vec{c} \right|^2 . . . . . \left( 2 \right)\]
From (1) and (2), we have
\[{AB}^2 + {AC}^2 = 2\left( {AD}^2 + {CD}^2 \right)\]
APPEARS IN
संबंधित प्रश्न
By vector method prove that the medians of a triangle are concurrent.
If point C `(barc)` divides the segment joining the points A(`bara`) and B(`barb`) internally in the ratio m : n, then prove that `barc=(mbarb+nbara)/(m+n)`
Show that the points A (1, –2, –8), B (5, 0, –2) and C (11, 3, 7) are collinear, and find the ratio in which B divides AC.
Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are `P(2veca + vecb)` and `Q(veca - 3vecb)` externally in the ratio 1: 2. Also, show that P is the mid point of the line segment RQ.
(Pythagoras's Theorem) Prove by vector method that in a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Prove by vector method that the sum of the squares of the diagonals of a parallelogram is equal to the sum of the squares of its sides.
Prove that the diagonals of a rectangle are perpendicular if and only if the rectangle is a square.
If the median to the base of a triangle is perpendicular to the base, then triangle is isosceles.
The position vector of points A and B are `6bar"a" + 2bar"b"` and `bar"a" - 3bar"b"`. If the point C divides AB in the ratio 3 : 2, show that the position vector of C is `3bar"a" - bar"b"`.
Prove that a quadrilateral is a parallelogram if and only if its diagonals bisect each other.
If two of the vertices of a triangle are A (3, 1, 4) and B (– 4, 5, –3) and the centroid of the triangle is at G (–1, 2, 1), then find the coordinates of the third vertex C of the triangle.
Prove that `(bar"a" xx bar"b").(bar"c" xx bar"d")` =
`|bar"a".bar"c" bar"b".bar"c"|`
`|bar"a".bar"d" bar"b".bar"d"|.`
If A(5, 1, p), B(1, q, p) and C(1, −2, 3) are vertices of triangle and `"G"("r", -4/3, 1/3)` is its centroid then find the values of p, q and r
Using vector method, find the incenter of the triangle whose vertices are A(0, 3, 0), B(0, 0, 4) and C(0, 3, 4)
P is the point of intersection of the diagonals of the parallelogram ABCD. If O is any point, then `overline"OA" + overline"OB" + overline"OC" + overline"OD"` = ______
If P(2, 2), Q(- 2, 4) and R(3, 4) are the vertices of Δ PQR then the equation of the median through vertex R is ______.
If the position vectors of points A and B are `hati + 8hatj + 4hatk` and `7hati + 2hatj - 8hatk`, then what will be the position vector of the midpoint of AB?
The image of the point (1, 6, 3) in the line `x/1 = (y - 1)/2 = (z - 2)/3` is ______
If the orthocentre and circumcentre of a triangle are (-3, 5, 1) and (6, 2, -2) respectively, then its centroid is ______
If M and N are the midpoints of the sides BC and CD respectively of a parallelogram ABCD, then `overline(AM) + overline(AN)` = ______
If `3bar"a" + 5bar"b" = 8bar"c"`, then A divides BC in tbe ratio ______.
The co-ordinates of the points which divides line segment joining the point A(2, –6, 8) and B(–1, 3,–4) internally in the ratio 1: 3' are ______.
Let `square`PQRS be a quadrilateral. If M and N are midpoints of the sides PQ and RS respectively then `bar"PS" + bar"OR"` = ______.
In ΔABC, P is the midpoint of BC, Q divides CA internally in the ratio 2:1 and R divides AB externally in the ratio 1:2, then ______.
Find the unit vector in the diret:tion of the vector `veca = hati + hatj + 2hatk`
What is the midpoint of the vector joining the point P(2, 3, 4) and Q(4, 1, –2)?
In ΔABC the mid-point of the sides AB, BC and CA are respectively (l, 0, 0), (0, m, 0) and (0, 0, n). Then, `("AB"^2 + "BC"^2 + "CA"^2)/("l"^2 + "m"^2 + "n"^2)` is equal to ______.
Using vector method, prove that the perpendicular bisectors of sides of a triangle are concurrent.
Let `A(bara)` and `B(barb)` be any two points in the space and `R(barr)` be the third point on the line AB dividing the segment AB externally in the ratio m : n, then prove that `barr = (mbarb - nbara)/(m - n)`.
The position vector of points A and B are `6bara + 2 barb and bara - 3 barb`. If point C divides AB in the ratio 3 : 2, then show that the position vector of C is `3bara - barb`.
The position vector of points A and B are `6 bara + 2 barb and bara - 3 barb`. If the point C divides AB in the ratio 3 : 2 then show that the position vector of C is `3 bara - barb`.
The position vector of points A and B are 6`bara + 2barb and bara - 3barb`. If the point C divides AB in the ratio 3 : 2 then show that the position vector of C is 3`bara - barb`.
