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.
Advertisements
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
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)`
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 \[{OP}^2 + {OQ}^2 = \frac{5}{9} {AB}^2\]
Prove using vectors: The quadrilateral obtained by joining mid-points of adjacent sides of a rectangle is a rhombus.
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.
Find the position vector of point R which divides the line joining the points P and Q whose position vectors are `2hati - hatj + 3hatk` and `- 5hati + 2hatj - 5hatk` in the ratio 3:2 is internally.
Find the position vector of point R which divides the line joining the points P and Q whose position vectors are `2hat"i" - hat"j" + 3hat"k"` and `- 5hat"i" + 2hat"j" - 5hat"k"` in the ratio 3 : 2 is externally.
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 the line segments joining the midpoints of the adjacent sides of a quadrilateral form a parallelogram.
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.
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.
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.
If `bara, barb` and `barc` are position vectors of the points A, B, C respectively and `5bara - 3barb - 2barc = bar0`, then find the ratio in which the point C divides the line segement BA.
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)
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 ______.
Let G be the centroid of a Δ ABC and O be any other point in that plane, then OA + OB + OC + CG = ?
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 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 ______.
If A, B, C are the vertices of a triangle whose position vectors are `overline("a"),overline("b"),overline("c")` and G is the centroid of the `triangle ABC,` then `overline("GA")+overline("GB")+overline("GC")` is ______.
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 ______.
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`
M and N are the mid-points of the diagonals AC and BD respectively of quadrilateral ABCD, then AB + AD + CB + CD is equal to ______.
If G(g), H(h) and (p) are centroid orthocentre and circumcentre of a triangle and xp + yh + zg = 0, then (x, y, z) is equal to ______.
The position vectors of three consecutive vertices of a parallelogram ABCD are `A(4hati + 2hatj - 6hatk), B(5hati - 3hatj + hatk)`, and `C(12hati + 4hatj + 5hatk)`. The position vector of D is given by ______.
If `bara, barb` and `barr` are position vectors of the points A, B and R respectively and R divides the line segment AB externally in the ratio m : n, then prove that `barr = (mbarb - nbara)/(m - n)`.
AB and CD are two chords of a circle intersecting at right angles to each other at P. If R is the centre of the circle, prove that:
`bar(PA) + bar(PB) + bar(PC) + bar(PD) = 2bar(PR)`
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 + 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.`
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`.
