Advertisements
Advertisements
Question
If points A (5, p) B (1, 5), C (2, 1) and D (6, 2) form a square ABCD, then p =
Options
7
3
6
8
Advertisements
Solution
The distance d between two points `(x_1 ,y_1) and (x_2 , y_2)` is given by the formula
` d= sqrt( (x_1 -x_2)^2 + (y_1 - y_2)^2)`
In a square all the sides are equal to each other.
Here the four points are A(5,p), B(1,5), C(2,1) and D(6,2).
The vertex ‘A’ should be equidistant from ‘B’ as well as ‘D’
Let us now find out the distances ‘AB’ and ‘AD’.
`AB = sqrt((5-1)^2 + (P -5)^2)`
`AB = sqrt((4)^2 + (p -5)^2)`
`AD = sqrt((5-6)^2 + (p-2)^2)`
`AD = sqrt((-1)^2 + (p-2)^2)`
These two need to be equal.
Equating the above two equations we have,
AB = AD
`sqrt((4)^2 +(p -5)^2 ) = sqrt((-1)^2 + (p-2)^2)`
Squaring on both sides we have,
`(4)^2 +(p -5)^2 = (-1)^2 + (p - 2)^2`
`16+p^2 + 25 - 10 p = 1 + p^2 + 4 - 4p`
6p = 36
p = 6
APPEARS IN
RELATED QUESTIONS
Two vertices of an isosceles triangle are (2, 0) and (2, 5). Find the third vertex if the length of the equal sides is 3.
Find the point on x-axis which is equidistant from the points (−2, 5) and (2,−3).
Find the coordinates of the point where the diagonals of the parallelogram formed by joining the points (-2, -1), (1, 0), (4, 3) and(1, 2) meet
The points (3, -4) and (-6, 2) are the extremities of a diagonal of a parallelogram. If the third vertex is (-1, -3). Find the coordinates of the fourth vertex.
Find the ratio in which the point (2, y) divides the line segment joining the points A (-2,2) and B (3, 7). Also, find the value of y.
Find the ratio in which the line segment joining (-2, -3) and (5, 6) is divided by x-axis Also, find the coordinates of the point of division in each case.
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.
In what ratio is the line segment joining A(2, -3) and B(5, 6) divide by the x-axis? Also, find the coordinates of the pint of division.
If the vertices of ΔABC be A(1, -3) B(4, p) and C(-9, 7) and its area is 15 square units, find the values of p
Find the ratio in which the point (−3, k) divides the line-segment joining the points (−5, −4) and (−2, 3). Also find the value of k ?
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 .
If the point P (m, 3) lies on the line segment joining the points \[A\left( - \frac{2}{5}, 6 \right)\] and B (2, 8), find the value of m.
Find the value of k, if the points A (8, 1) B(3, −4) and C(2, k) are collinear.
If the points A(−1, −4), B(b, c) and C(5, −1) are collinear and 2b + c = 4, find the values of b and c.
If the distance between the points (3, 0) and (0, y) is 5 units and y is positive. then what is the value of y?
A line intersects the y-axis and x-axis at P and Q , respectively. If (2,-5) is the mid-point of PQ, then the coordinates of P and Q are, respectively
Find the point on the y-axis which is equidistant from the points (S, - 2) and (- 3, 2).
The line 3x + y – 9 = 0 divides the line joining the points (1, 3) and (2, 7) internally in the ratio ______.
The coordinates of the point where the line 2y = 4x + 5 crosses x-axis is ______.
Ryan, from a very young age, was fascinated by the twinkling of stars and the vastness of space. He always dreamt of becoming an astronaut one day. So, he started to sketch his own rocket designs on the graph sheet. One such design is given below :
Based on the above, answer the following questions:
i. Find the mid-point of the segment joining F and G. (1)
ii. a. What is the distance between the points A and C? (2)
OR
b. Find the coordinates of the points which divides the line segment joining the points A and B in the ratio 1 : 3 internally. (2)
iii. What are the coordinates of the point D? (1)
