Advertisements
Advertisements
प्रश्न
Show that P(–2, 2), Q(2, 2) and R(2, 7) are vertices of a right angled triangle.
Advertisements
उत्तर
Distance between two points = `sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2`
By distance formula,
PQ = `sqrt([2 - (-2)]^2 + (2 - 2)^2`
= `sqrt((2 + 2)^2 + (0)^2`
= `sqrt((4)^2`
= 4 ...(i)
QR = `sqrt((2 - 2)^2 + (7 - 2)^2`
= `sqrt((0)^2 + (5)^2`
= `sqrt((5)^2`
= 5 ...(ii)
PR = `sqrt([2 -(-2)]^2 + (7 - 2)^2`
= `sqrt((2 + 2)^2 + (5)^2`
= `sqrt((4)^2 + (5)^2`
= `sqrt(16 + 25)`
= `sqrt(41)`
Now, PR2 = `(sqrt(41))^2`
= 41 ...(iii)
Consider, PQ2 + QR2
= 42 + 52
= 16 + 25
= 41 ...[From (i) and (ii)]
∴ PR2 = PQ2 + QR2 ...[From (iii)]
∴ ∆PQR is a right angled triangle. ...[Converse of Pythagoras theorem]
∴ Points P, Q, and R are the vertices of a right angled triangle.
APPEARS IN
संबंधित प्रश्न
If the point P(2, 2) is equidistant from the points A(–2, k) and B(–2k, –3), find k. Also, find the length of AP.
Show that the points (a, a), (–a, –a) and (– √3 a, √3 a) are the vertices of an equilateral triangle. Also find its area.
Find the distance between the points (0, 0) and (36, 15). Can you now find the distance between the two towns A and B discussed in Section 7.2.
Name the type of quadrilateral formed, if any, by the following point, and give reasons for your answer:
(4, 5), (7, 6), (4, 3), (1, 2)
Find the distance between the following pair of points:
(a sin α, –b cos α) and (–a cos α, b sin α)
Find the circumcenter of the triangle whose vertices are (–2, –3), (–1, 0), (7, –6).
Given a triangle ABC in which A = (4, −4), B = (0, 5) and C = (5, 10). A point P lies on BC such that BP : PC = 3 : 2. Find the length of line segment AP.
Find the distance between the following pairs of point.
W `((- 7)/2 , 4)`, X (11, 4)
Prove that the following set of point is collinear :
(4, -5),(1 , 1),(-2 , 7)
Find the coordinate of O , the centre of a circle passing through P (3 , 0), Q (2 , `sqrt 5`) and R (`-2 sqrt 2` , -1). Also find its radius.
Find the coordinates of O, the centre passing through A( -2, -3), B(-1, 0) and C(7, 6). Also, find its radius.
Prove taht the points (-2 , 1) , (-1 , 4) and (0 , 3) are the vertices of a right - angled triangle.
ABCD is a square . If the coordinates of A and C are (5 , 4) and (-1 , 6) ; find the coordinates of B and D.
Given A = (x + 2, -2) and B (11, 6). Find x if AB = 17.
KM is a straight line of 13 units If K has the coordinate (2, 5) and M has the coordinates (x, – 7) find the possible value of x.
Show that the point (0, 9) is equidistant from the points (–4, 1) and (4, 1).
Points A(4, 3), B(6, 4), C(5, –6) and D(–3, 5) are the vertices of a parallelogram.
The point A(2, 7) lies on the perpendicular bisector of line segment joining the points P(6, 5) and Q(0, – 4).
The point P(–2, 4) lies on a circle of radius 6 and centre C(3, 5).
|
Case Study Trigonometry in the form of triangulation forms the basis of navigation, whether it is by land, sea or air. GPS a radio navigation system helps to locate our position on earth with the help of satellites. |
- Make a labelled figure on the basis of the given information and calculate the distance of the boat from the foot of the observation tower.
- After 10 minutes, the guard observed that the boat was approaching the tower and its distance from tower is reduced by 240(`sqrt(3)` - 1) m. He immediately raised the alarm. What was the new angle of depression of the boat from the top of the observation tower?
