Advertisements
Advertisements
Question
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
Solution
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
RELATED QUESTIONS
By vector method prove that the medians of a triangle are concurrent.
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.
(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.
If AD is the median of ∆ABC, using vectors, prove that \[{AB}^2 + {AC}^2 = 2\left( {AD}^2 + {CD}^2 \right)\]
Let `A(bara)` and `B(barb)` are any two points in the space and `"R"(bar"r")` be a point on the line segment AB dividing it internally in the ratio m : n, then prove that `barr = (mbarb + nbara)/(m + n)`.
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.
If the points A(3, 0, p), B(–1, q, 3) and C(–3, 3, 0) are collinear, then find
- the ratio in which the point C divides the line segment AB
- the values of p and q.
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.
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.
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.
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"`.
Find the volume of a parallelopiped whose coterimus edges are represented by the vectors `hat"i" + hat"k", hat"i" + hat"k", hat"i" + hat"j"`. Also find volume of tetrahedron having these coterminus edges.
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.
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 quadrilateral ABCD, M and N are the mid-points of the sides AB and CD respectively. If AD + BC = tMN, then t = ____________.
In a triangle ABC, if `1/(a + c) + 1/(b + c) = 3/(a + b + c)` then angle C is equal to ______
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 G and G' are the centroids of the triangles ABC and A'B'C', then `overline("A""A"^') + overline("B""B"^') + overline("C""C"^')` is equal to ______
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`
If D, E, F are the mid points of the sides BC, CA and AB respectively of a triangle ABC and 'O' is any point, then, `|vec(AD) + vec(BE) + vec(CF)|`, is ______.
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 `6bara + 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 `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 3`bara - barb`.
The position vectors of points A and B are 6`bara` + 2`barb` 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 - b`.
