Latur School Exam Papers 2026
Raja Narayanlal Lahoti English School
Class 9 | Term Test-I
37 Drone Navigation (Coordinate Geometry)
A drone is tested at three points P (2,3), Q (8,3), and R (2,9). Engineers want to verify if the path forms a right-angled triangle for optimal navigation. The longest side will act as the drone's main route, and a charging station will be placed at the midpoint of this side. Using coordinate geometry concepts, answer the following questions.[cite: 49]
(A) Verify whether the triangle is right-angled. (2 Marks)
(B) Determine which side is the hypotenuse. (1 Mark)
(C) Find the midpoint of the hypotenuse. (1 Mark)
(B) Determine which side is the hypotenuse. (1 Mark)
(C) Find the midpoint of the hypotenuse. (1 Mark)
OR
(C) Find coordinates of charging station. (1 Mark)
Concept:
1
Distance Formula: \( d^2 = (x_2 - x_1)^2 + (y_2 - y_1)^2 \). We can use this to find the squares of all three sides.
2
Converse of Pythagoras: If the sum of the squares of the two smaller sides equals the square of the longest side (\(a^2 + b^2 = c^2\)), the triangle is perfectly right-angled.
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Hint:
1
Find \(PQ^2\), \(PR^2\), and \(QR^2\) using the coordinates given. You will see that two of them are equal to 36, and the third is 72. Does \(36 + 36 = 72\)?
2
For part (C) and the OR choice, they are actually asking the exact same question! The charging station is placed at the midpoint of the longest route (hypotenuse). Use the midpoint formula on points Q and R.
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Mistakes to Avoid:
!
Skipping Distance Formula: Just "looking" at the coordinates and saying it's a right triangle because P and Q share a y-coordinate, and P and R share an x-coordinate. While true visually, a formal verification requires algebraic proof via Pythagoras!
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Given Points: \( P(2,3) \), \( Q(8,3) \), \( R(2,9) \)
PART (A) SOLUTION:
Calculate the squares of distances between the points:
\( PQ^2 = (8 - 2)^2 + (3 - 3)^2 = 6^2 + 0^2 = 36 \) [½ M]
\( PR^2 = (2 - 2)^2 + (9 - 3)^2 = 0^2 + 6^2 = 36 \) [½ M]
\( QR^2 = (2 - 8)^2 + (9 - 3)^2 = (-6)^2 + 6^2 = 36 + 36 = 72 \) [½ M]
Now, observe the relation:
\( PQ^2 + PR^2 = 36 + 36 = 72 \)
Since \( PQ^2 + PR^2 = QR^2 \), by the converse of Pythagoras theorem, \( \Delta PQR \) is a right-angled triangle (right-angled at P). [½ M]
Verified: It is a right-angled triangle.
PART (B) SOLUTION:
The hypotenuse is the longest side of a right-angled triangle.
Since \( QR^2 = 72 \) is greater than \( PQ^2 \) and \( PR^2 \), side QR is the longest.
Therefore, the hypotenuse is QR. [1 M]
Hypotenuse = Side QR
PART (C) / OR (C) SOLUTION:
(Note: Both questions ask for the same calculation, as the charging station is placed exactly at the midpoint of the hypotenuse.)
The hypotenuse connects \( Q(8,3) \) and \( R(2,9) \).
Using the midpoint formula: \( M(x, y) = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \) [½ M]
Midpoint = \( \left( \frac{8 + 2}{2}, \frac{3 + 9}{2} \right) \)
Midpoint = \( \left( \frac{10}{2}, \frac{12}{2} \right) \)
Midpoint = \( (5, 6) \) [½ M]
Coordinates = (5, 6)
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