Advertisements
Advertisements
प्रश्न
In a quadrilateral ABCD, prove that \[{AB}^2 + {BC}^2 + {CD}^2 + {DA}^2 = {AC}^2 + {BD}^2 + 4 {PQ}^2\] where P and Q are middle points of diagonals AC and BD.
उत्तर
Let ABCD be the quadrilateral. Taking A as the origin, let the position vectors of B, C and D be \[\vec{b} , \vec{c}\] and \[\vec{d}\] respectively.
Then,
Position vector of P =\[\frac{\vec{c}}{2}\]........(Mid-point formula)
Position vector of Q = \[\frac{\vec{b} + \vec{d}}{2}\]..............(Mid-point formula)
Now,
\[{AB}^2 + {BC}^2 + {CD}^2 + {DA}^2 \]
\[ = \left| \vec{AB} \right|^2 + \left| \vec{BC} \right|^2 + \left| \vec{CD} \right|^2 + \left| \vec{DA} \right|^2 \]
\[ = \left| \vec{b} \right|^2 + \left| \vec{c} - \vec{b} \right|^2 + \left| \vec{d} - \vec{c} \right|^2 + \left| \vec{d} \right|^2 \]
\[ = \left| \vec{b} \right|^2 + \left| \vec{c} \right|^2 - 2 \vec{c} . \vec{b} + \left| \vec{b} \right|^2 + \left| \vec{d} \right|^2 - 2 \vec{d} . \vec{c} + \left| \vec{c} \right|^2 + \left| \vec{d} \right|\]
\[ = 2 \left| \vec{b} \right|^2 + 2 \left| \vec{c} \right|^2 + 2 \left| \vec{d} \right|^2 - 2 \vec{b} . \vec{c} - 2 \vec{c} . \vec{d} . . . . . \left( 1 \right)\]
Also,
\[{AC}^2 + {BD}^2 + 4 {PQ}^2 \]
\[ = \left| \vec{AC} \right|^2 + \left| \vec{BD} \right|^2 + 4 \left| \vec{PQ} \right|^2 \]
\[ = \left| \vec{c} \right|^2 + \left| \vec{d} - \vec{b} \right|^2 + 4 \left| \frac{\vec{b} + \vec{d}}{2} - \frac{\vec{c}}{2} \right|^2 \]
\[ = \left| \vec{c} \right|^2 + \left| \vec{d} - \vec{b} \right|^2 + \left| \vec{b} + \vec{d} \right|^2 - 2\left( \vec{b} + \vec{d} \right) . \vec{c} + \left| \vec{c} \right|^2 \]
\[ = 2 \left| \vec{c} \right|^2 + 2 \left| \vec{d} \right|^2 + 2 \left| \vec{b} \right|^2 - 2 \vec{b} . \vec{c} - 2 \vec{d} . \vec{c} \]
\[ = 2 \left| \vec{b} \right|^2 + 2 \left| \vec{c} \right|^2 + 2 \left| \vec{d} \right|^2 - 2 \vec{b} . \vec{c} - 2 \vec{c} . \vec{d} . . . . . \left( 2 \right)\]
From (1) and (2), we have
\[{AB}^2 + {BC}^2 + {CD}^2 + {DA}^2 = {AC}^2 + {BD}^2 + 4 {PQ}^2\]
APPEARS IN
संबंधित प्रश्न
If `bar p = hat i - 2 hat j + hat k and bar q = hat i + 4 hat j - 2 hat k` are position vector (P.V.) of points P and Q, find the position vector of the point R which divides segment PQ internally in the ratio 2:1
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.
If the origin is the centroid of the triangle whose vertices are A(2, p, –3), B(q, –2, 5) and C(–5, 1, r), then find the values of p, q, r.
In a triangle OAB,\[\angle\]AOB = 90º. If P and Q are points of trisection of AB, prove that \[{OP}^2 + {OQ}^2 = \frac{5}{9} {AB}^2\]
(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 that the diagonals of a rectangle are perpendicular if and only if the rectangle is a square.
Find the position vector of midpoint M joining the points L(7, –6, 12) and N(5, 4, –2).
Prove that the median of a trapezium is parallel to the parallel sides of the trapezium and its length is half of the sum of the lengths of the parallel sides.
In Δ OAB, E is the midpoint of OB and D is the point on AB such that AD : DB = 2 : 1. If OD and AE intersect at P, then determine the ratio OP : PD using vector methods.
If the centroid of a tetrahedron OABC is (1, 2, - 1) where A(a, 2, 3), B(1, b, 2), C(2, 1, c), find the distance of P(a, b, c) from origin.
Find the centroid of tetrahedron with vertices K(5, −7, 0), L(1, 5, 3), M(4, −6, 3), N(6, −4, 2)
The points A, B, C have position vectors `bar"a", bar"b" and bar"c"` respectively. The point P is the midpoint of AB. Find the vector `bar"PC"` in terms of `bar"a", bar"b", bar"c"`.
If D, E, F are the midpoints of the sides BC, CA, AB of a triangle ABC, prove that `bar"AD" + bar"BE" + bar"CF" = bar0`.
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 G(a, 2, −1) is the centroid of the triangle with vertices P(1, 2, 3), Q(3, b, −4) and R(5, 1, c) then find the values of a, b and c
Prove that medians of a triangle are concurrent
Prove that altitudes of a triangle are concurrent
Prove that the angle bisectors of a triangle are concurrent
If A(1, 3, 2), B(a, b, - 4) and C(5, 1, c) are the vertices of triangle ABC and G(3, b, c) is its centroid, then
If the plane 2x + 3y + 5z = 1 intersects the co-ordinate axes at the points A, B, C, then the centroid of Δ ABC is ______.
In a triangle ABC, if `1/(a + c) + 1/(b + c) = 3/(a + b + c)` then angle C is equal to ______
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 ______.
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"` = ______.
What is the midpoint of the vector joining the point P(2, 3, 4) and Q(4, 1, –2)?
The position vector of points A and B are `6 bar "a" + 2 bar "b" and bar "a" - 3 bar"b"`. If the point C divided AB in the ratio 3 : 2, show that the position vector of C is `3 bar "a" - bar "b".`
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 `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 `3bara - barb.`
