Advertisements
Advertisements
प्रश्न
The base BC of an equilateral triangle ABC lies on y-axis. The coordinates of point C are (0, −3). The origin is the midpoint of the base. Find the coordinates of the points A and B. Also, find the coordinates of another point D such that ABCD is a rhombus.
Advertisements
उत्तर
Given:
ΔABC is equilateral
Base BC lies on the y–axis
C = (0, –3)
Origin (0, 0) is midpoint of BC
B = (0, b), C = (0, −3)
`0 = (b+(-3))/2`
b − 3 = 0
b = 3
B = (0, 3)
C = (0, –3)
In an equilateral triangle,
height = `sqrt3/2 xx "side"`
Height = `sqrt3/2 xx 6`
`= 3sqrt3`
Base lies on y-axis, so the altitude is horizontal, meaning A lies on the x-axis but at distance `3sqrt3` from O.
`A = (+-3sqrt3, 0)`
A1 = `(3sqrt3, 0)`
A2 = (`-3sqrt3, 0`)
A rhombus has all sides equal, and ABCD is cyclic order.
A = (`3sqrt3`, 0), B = (0, 3), C = (0, −3)
To form a rhombus ABCD, the 4th point D is:
D = A + C − B
D = (`3sqrt3`, 0) + (0, −3) − (0, 3)
D = (`3sqrt3`, −6)
Base points:
B = (0, 3), C = (0, −3)
Vertices of equilateral triangle:
A = (`3sqrt3`, 0) or A = (`−3sqrt3`, 0)
Point D such that ABCD is a rhombus:
If A = (`3sqrt3`, 0):
D = (`3sqrt3`, −6)
If A = (`−3sqrt3`, 0):
D = (`-3sqrt3`, −6)
संबंधित प्रश्न
Find a point on the x-axis which is equidistant from the points (7, 6) and (−3, 4).
In the seating arrangement of desks in a classroom three students Rohini, Sandhya and Bina are seated at A(3, 1), B(6, 4), and C(8, 6). Do you think they are seated in a line?
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.
If the poin A(0,2) is equidistant form the points B (3, p) and C (p ,5) find the value of p. Also, find the length of AB.
If the points p (x, y) is point equidistant from the points A (5, 1)and B (–1, 5), Prove that 3x = 2y
Show that the points A(2,1), B(5,2), C(6,4) and D(3,3) are the angular points of a parallelogram. Is this figure a rectangle?
If the point C(k,4) divides the join of A(2,6) and B(5,1) in the ratio 2:3 then find the value of k.
Find the centroid of ΔABC whose vertices are A(2,2) , B (-4,-4) and C (5,-8).
In what ratio does the point C (4,5) divides the join of A (2,3) and B (7,8) ?
A line segment is of length 10 units. If the coordinates of its one end are (2, −3) and the abscissa of the other end is 10, then its ordinate is
The area of the triangle formed by (a, b + c), (b, c + a) and (c, a + b)
The ratio in which (4, 5) divides the join of (2, 3) and (7, 8) is
Write the equations of the x-axis and y-axis.
Find the coordinates of the point of intersection of the graph of the equation x = 2 and y = –3.
The distance of the point P(2, 3) from the x-axis is ______.
Point (–3, 5) lies in the ______.
Point (0, –7) lies ______.
Ordinate of all points on the 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)
