Advertisements
Advertisements
प्रश्न
Prove using vectors: The quadrilateral obtained by joining mid-points of adjacent sides of a rectangle is a rhombus.
Advertisements
उत्तर
ABCD is a rectangle. Let P, Q, R and S be the mid-points of the sides AB, BC, CD and DA, respectively.
Now,
\[\vec{PQ} = \vec{PB} + \vec{BQ} = \frac{1}{2} \vec{AB} + \frac{1}{2} \vec{BC} = \frac{1}{2}\left( \vec{AB} + \vec{BC} \right) = \frac{1}{2} \vec{AC}\]..............( 1 )
\[\vec{SR} = \vec{SD} + \vec{DR} = \frac{1}{2} \vec{AD} + \frac{1}{2} \vec{DC} = \frac{1}{2}\left( \vec{AD} + \vec{DC} \right) = \frac{1}{2} \vec{AC}\] ...............( 2 )
From (1) and (2), we have
\[\vec{PQ} = \vec{SR}\]
So, the sides PQ and SR are equal and parallel. Thus, PQRS is a parallelogram.
Now,
\[\left| \vec{PQ} \right|^2 = \vec{PQ} . \vec{PQ} \]
\[ \Rightarrow \left| \vec{PQ} \right|^2 = \left( \vec{PB} + \vec{BQ} \right) . \left( \vec{PB} + \vec{BQ} \right)\]
\[ \Rightarrow \left| \vec{PQ} \right|^2 = \left| \vec{PB} \right|^2 + 2 \vec{PB} . \vec{BQ} + \left| \vec{BQ} \right|^2 \]
\[ \Rightarrow \left| \vec{PQ} \right|^2 = \left| \vec{PB} \right|^2 + 0 + \left| \vec{BQ} \right|^2 \left( \vec{PB} \perp \vec{BQ} \right)\]
\[ \Rightarrow \left| \vec{PQ} \right|^2 = \left| \vec{PB} \right|^2 + \left| \vec{BQ} \right|^2 ....................\left( 3 \right)\]
Also,
\[\left| \vec{PS} \right|^2 = \vec{PS} . \vec{PS} \]
\[ \Rightarrow \left| \vec{PS} \right|^2 = \left( \vec{PA} + \vec{AS} \right) . \left( \vec{PA} + \vec{AS} \right)\]
\[ \Rightarrow \left| \vec{PS} \right|^2 = \left| \vec{PA} \right|^2 + 2 \vec{PA} . \vec{AS} + \left| \vec{AS} \right|^2 \]
\[ \Rightarrow \left| \vec{PS} \right|^2 = \left| \vec{PB} \right|^2 + 0 + \left| \vec{BQ} \right|^2 \left( \vec{PA} \perp \vec{AS} \right)\]
\[ \Rightarrow \left| \vec{PS} \right|^2 = \left| \vec{PB} \right|^2 + \left| \vec{BQ} \right|^2........................\left( 4 \right)\]
From (3) and (4), we have
\[\left| \vec{PQ} \right|^2 = \left| \vec{PS} \right|^2 \]
\[ \Rightarrow \left| \vec{PQ} \right| = \left| \vec{PS} \right|\]
So, the adjacent sides of the parallelogram are equal. Hence, PQRS is a rhombus.
APPEARS IN
संबंधित प्रश्न
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.
In a triangle OAB,\[\angle\]AOB = 90º. If P and Q are points of trisection of AB, prove that \[{OP}^2 + {OQ}^2 = \frac{5}{9} {AB}^2\]
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.
Find the position vector of midpoint M joining the points L(7, –6, 12) and N(5, 4, –2).
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 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 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 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 = ?
In a quadrilateral ABCD, M and N are the mid-points of the sides AB and CD respectively. If AD + BC = tMN, then t = ____________.
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 M and N are the midpoints of the sides BC and CD respectively of a parallelogram ABCD, then `overline(AM) + overline(AN)` = ______
If G`(overlineg)` is the centroid, `H(overlineh)` is the orthocentre and P`(overlinep)` is the circumcentre of a triangle and `xoverlinep + yoverlineh + zoverlineg = 0`, then ______
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 ______.
Let `square`PQRS be a quadrilateral. If M and N are midpoints of the sides PQ and RS respectively then `bar"PS" + bar"OR"` = ______.
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)?
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 ______.
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 ______.
ΔABC has vertices at A = (2, 3, 5), B = (–1, 3, 2) and C = (λ, 5, µ). If the median through A is equally inclined to the axes, then the values of λ and µ respectively are ______.
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 ______.
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)`
If `bara, barb, barc` are the position vectors of the points A, B, C respectively and `5 bar a - 3 bar b - 2 bar c = bar 0`, then find the ratio in which the point C divides the line segment BA.
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`.
