Visual Lens | Coordinate Geometry | MEDIUM
The (x, y) coordinates of vertices P, Q and R of a parallelogram PQRS are (-3, -2), (1, -5) and (9, 1), respectively. If the diagonal SQ intersects the x-axis at (a, 0), then the value of a is
- 10/3
- 27/7
- 13/4
- 29/9
Answer
For a parallelogram P-Q-R-S in that order, S = P + R − Q.
P = (−3, −2), Q = (1, −5), R = (9, 1) ⇒ S = (−3+9−1, −2+1−(−5)) = (5, 4).
Line SQ passes through Q(1, −5) and S(5, 4). Slope = (4 − (−5)) / (5 − 1) = 9/4.
Equation: y + 5 = (9/4)(x − 1). Put y = 0 to find x-intercept: 5 = (9/4)(x − 1).
x − 1 = 20/9 ⇒ x = 1 + 20/9 = 29/9.
Answer: 29/9 (option 4).
Visual Lens | Odd even numbers
In the set of consecutive odd numbers {1, 3, 5, …, 57}, there is a number k such that the sum of all the elements less than k is equal to the sum of all the elements greater than k. Then, k equals
- 37
- 43
- 39
- 41
Answer
Let the diagonals be d1 and d2.
Area of a rhombus = 1/2 × d1 × d2 = 396 → d1 × d2 = 792.
In a rhombus, diagonals bisect each other at right angles.
So by Pythagoras: (d1/2)^2 + (d2/2)^2 = 36^2.
Multiply by 4: d1^2 + d2^2 = 4 × 1296 = 5184.
Use identity: (d1 − d2)^2 = (d1^2 + d2^2) − 2(d1 × d2).
Substitute: (d1 − d2)^2 = 5184 − 2×792 = 5184 − 1584 = 3600.
So |d1 − d2| = √3600 = 60.
Final Answer: 60
S
Answer
dd
