Advertisements
Advertisements
Question
ABCD is rectangle formed by the points A(-1, -1), B(-1, 4), C(5, 4) and D(5, -1). If P,Q,R and S be the midpoints of AB, BC, CD and DA respectively, Show that PQRS is a rhombus.
Advertisements
Solution
Here, the points P ,Q, Rand S are the midpoint of ,AB ,BC, CD and DA respectively. Then
`"Coordinates of P = ((-1-1)/2 , (-1+4)/2) = (-1,3/2)`
`"Coordinates of Q = ((-1+5)/2 , (4+4)/2) = (2,.4)`
`"Coordinates of R = ((5+5)/2 , (4-1)/2)= (5,3/2)`
`"Coordinates of " S = ((-1+5)/2 ,(-1-1)/2) = (2,-1)`
Now,
`PQ = sqrt((2+1)^2 +(4-3/2)^2) = sqrt(9+25/4) = sqrt(61/2)`
`QR = sqrt((5-2)^2 +(3/2-4)^2 )= sqrt(9+25/4) = sqrt(61/2)`
`RS = sqrt((5-2)^2 +(3/2+1)^2 )= sqrt(9+25/4) = sqrt(61/2)`
`SP = sqrt((2+1)^2 +(-1-3/2)^2 )= sqrt(9+25/4) = sqrt(61/2)`
` PR = sqrt((5-1)^2 +(3/2-3/2)^2) = sqrt(36) = 6`
`QS = sqrt((2-2)^2 +(-1-4)^2) = sqrt(25) =5`
Thus, PQ = QR = RS = SP and PR ≠ QS therefore PQRS is a rhombus
APPEARS IN
RELATED QUESTIONS
If the points A(k + 1, 2k), B(3k, 2k + 3) and C(5k − 1, 5k) are collinear, then find the value of k
On which axis do the following points lie?
P(5, 0)
Let ABCD be a square of side 2a. Find the coordinates of the vertices of this square when A coincides with the origin and AB and AD are along OX and OY respectively.
Find the coordinates of the circumcentre of the triangle whose vertices are (3, 0), (-1, -6) and (4, -1). Also, find its circumradius.
Show that the points A(5, 6), B(1, 5), C(2, 1) and D(6,2) are the vertices of a square.
Find the points on the x-axis, each of which is at a distance of 10 units from the point A(11, –8).
Find the ratio in which the point P(m, 6) divides the join of A(-4, 3) and B(2, 8) Also, find the value of m.
The ordinate of any point on x-axis is
The abscissa of a point is positive in the
If the point \[C \left( - 1, 2 \right)\] divides internally the line segment joining the points A (2, 5) and B( x, y ) in the ratio 3 : 4 , find the value of x2 + y2 .
Find the value of k if points A(k, 3), B(6, −2) and C(−3, 4) are collinear.
\[A\left( 6, 1 \right) , B(8, 2) \text{ and } C(9, 4)\] are three vertices of a parallelogram ABCD . If E is the mid-point of DC , find the area of \[∆\] ADE.
Find the value of a so that the point (3, a) lies on the line represented by 2x − 3y + 5 = 0
What is the distance between the points A (c, 0) and B (0, −c)?
If A (2, 2), B (−4, −4) and C (5, −8) are the vertices of a triangle, than the length of the median through vertex C is
If Points (1, 2) (−5, 6) and (a, −2) are collinear, then a =
If the line segment joining the points (3, −4), and (1, 2) is trisected at points P (a, −2) and Q \[\left( \frac{5}{3}, b \right)\] , Then,
What are the coordinates of origin?
A tiling or tessellation of a flat surface is the covering of a plane using one or more geometric shapes, called tiles, with no overlaps and no gaps. Historically, tessellations were used in ancient Rome and in Islamic art. You may find tessellation patterns on floors, walls, paintings etc. Shown below is a tiled floor in the archaeological Museum of Seville, made using squares, triangles and hexagons.
A craftsman thought of making a floor pattern after being inspired by the above design. To ensure accuracy in his work, he made the pattern on the Cartesian plane. He used regular octagons, squares and triangles for his floor tessellation pattern
Use the above figure to answer the questions that follow:
- What is the length of the line segment joining points B and F?
- The centre ‘Z’ of the figure will be the point of intersection of the diagonals of quadrilateral WXOP. Then what are the coordinates of Z?
- What are the coordinates of the point on y-axis equidistant from A and G?
OR
What is the area of Trapezium AFGH?
