Advertisements
Advertisements
प्रश्न
Let P(11, 7), Q(13.5, 4) and R(9.5, 4) be the midpoints of the sides AB, BC and AC respectively of ∆ABC. Find the coordinates of the vertices A, B and C. Hence find the area of ∆ABC and compare this with area of ∆PQR.
Advertisements
उत्तर
Let the vertices of the ∆ABC be A(x1, y1), B(x2, y2), C(x3, y3)
Mid point of AB = `((x_1 + x_2)/2, (y_1 + y_2)/2)`
(11, 7) = `((x_1 + x_2)/2, (y_1 + y_2)/2)`
|
`(x_1 + x_2)/2` = 11 x1 + x2 = 22 ...(1) |
`(y_1 + y_2)/2` = 7 y1 + y2 = 14 ...(2) |
Mid point of BC = `((x_2 + x_3)/2, (y_2 + y_3)/2)`
⇒ (13.5, 4) = `((x_2 + x_3)/2, (y_2 + y_3)/2)`
|
`(x_2 + x_3)/2` = 13.5 x2 + x3 = 27 ...(3) |
`(y_2 + y_3)/2` = 4 y2 + y3 = 8 ...(4) |
Mid point of AC = `((x_1 + x_3)/2, (y_1 + y_3)/2)`
(9.5, 4) = `((x_1 + x_2)/2, (y_1 + y_3)/2)`
|
`(x_1 + x_3)/2` = 9.5 x1 + x3 = 19 ...(5) |
`(y_1 + y_3)/2` = 4 y1 + y3 = 8 ...(6) |
Add (1), (3) and (5)
2x1 + 2x2 + 2x3 = 22 + 27 + 19
2(x1 + x2 + x3) = 68
x1 + x2 + x3 = 34
From (1) ⇒ x1 + x2 = 22
x3 = 34 – 22 = 12
From (3) ⇒ x2 + x3 = 27
x1 = 34 – 27 = 7
From (5) ⇒ x1 + x3 = 19
x2 = 34 – 19 = 15
Add (2), (4) and (6)
2y1 + 2y2 + 2y3 = 14 + 8 + 8
2(y1 + y2 + y3) = 30
y1 + y2 + y3 = 15
From (2) ⇒ y1 + y2 = 14
y3 = 15 – 14 = 1
From (4) ⇒ y2 + y3 = 18
y1 = 15 – 8 = 7
From (6) ⇒ y1 + y3 = 8
y2 = 15 – 8 = 7
The vertices of a ΔABC are A(7, 7), B(15, 7) and C(12, 1)
Area of ΔABC = `1/2[(x_1y_2 + x_2y_3 + x_3y_1) - (x_2y_1 + x_3y_2 + x_1y_3)]`
= `1/2[(7 + 84 + 105) - (84 + 15 + 49)]`
= `1/2[196 - 148]`
= `1/2 xx 48`
= 24 sq. units
Area of ΔPRQ = `1/2[(44 + 8 + 94.5) - (66.5 + 54 + 44)]`
= `1/2[176.5 - 164.5]`
= `1/2 xx 12`
= 6 sq. units
APPEARS IN
संबंधित प्रश्न
The distance between parallel sides of a trapezium is 15 cm and the length of the line segment joining the mid-points of its non-parallel sides is 26 cm. Find the area of the trapezium.
Find the area of a quadrilateral one of whose diagonals is 30 cm long and the perpendiculars from the other two vertices are 19 cm and 11 cm respectively.
The diagonals of a quadrilateral are 16 cm and 13 cm. If they intersect each other at right angles; find the area of the quadrilateral.
The length and the breadth of a rectangle are 6 cm and 4 cm respectively. Find the height of a triangle whose base is 6 cm and the area is 3 times that of the rectangle.
The cost of enclosing a rectangular garden with a fence all around, at the rate of 75 paise per metre, is Rs. 300. If the length of the garden is 120 metres, find the area of the field in square metres.
The area of a rhombus is 216 sq. cm. If it's one diagonal is 24 cm; find:
(i) Length of its other diagonal,
(ii) Length of its side,
(iii) The perimeter of the rhombus.
Two adjacent sides of a parallelogram are 28 cm and 26 cm. If one diagonal of it is 30 cm long; find the area of the parallelogram. Also, find the distance between its shorter sides.
Find the diagonal of a quadrilateral whose area is 756cm2 and the perpendicular from the opposite vertices are 17cm and 19cm.
PQRS is a rectangle formed by joining the points P(– 1, – 1), Q(– 1, 4), R(5, 4) and S(5, – 1). A, B, C and D are the mid-points of PQ, QR, RS and SP respectively. Is the quadrilateral ABCD a square, a rectangle or a rhombus? Justify your answer.
If the diagonal d of a quadrilateral is doubled and the heights h1 and h2 falling on d are halved, then the area of quadrilateral is ______.
