Question

If the graph of the equation $ 4x + 3y = 12 $ cuts the coordinate axes at $ A $ and $ B $, then the hypotenuse of right triangle $ \triangle AOB $ is of length:

• A. $ 4 $ units
• B. $ 3 $ units
• C. $ 5 $ units
• D. None of these

Hint:

To find the hypotenuse of a right triangle formed by the intercepts of a linear equation, first determine the intercepts, then use the Pythagorean theorem or the distance formula to calculate the hypotenuse.

Answer 1

The Correct Option is C

Step 1: Find the Points of Intersection with the Coordinate Axes.
The given equation is: \[ 4x + 3y = 12. \] Intersection with the x-axis (\( y = 0 \)):
Substitute \( y = 0 \) into the equation: \[ 4x + 3(0) = 12 \implies 4x = 12 \implies x = 3. \] So, the point of intersection with the x-axis is \( A(3, 0) \). Intersection with the y-axis (\( x = 0 \)):
Substitute \( x = 0 \) into the equation: \[ 4(0) + 3y = 12 \implies 3y = 12 \implies y = 4. \] So, the point of intersection with the y-axis is \( B(0, 4) \). Step 2: Determine the Lengths of the Legs of the Right Triangle.
The points \( A(3, 0) \) and \( B(0, 4) \) form a right triangle \( \triangle AOB \) with the origin \( O(0, 0) \). The lengths of the legs of the triangle are: Distance from \( O \) to \( A \) (along the x-axis): \[ OA = 3 \text{ units}. \] Distance from \( O \) to \( B \) (along the y-axis): \[ OB = 4 \text{ units}. \] Step 3: Calculate the Hypotenuse.
The hypotenuse \( AB \) is the distance between points \( A(3, 0) \) and \( B(0, 4) \). Using the distance formula: \[ AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}. \] Substitute \( A(3, 0) \) and \( B(0, 4) \): \[ AB = \sqrt{(0 - 3)^2 + (4 - 0)^2} = \sqrt{(-3)^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5. \] Step 4: Analyze the Options.
Option (1): \( 4 \) units — Incorrect, as the hypotenuse is \( 5 \) units.
Option (2): \( 3 \) units — Incorrect, as the hypotenuse is \( 5 \) units.
Option (3): \( 5 \) units — Correct, as it matches the calculated value.
Option (4): None of these — Incorrect, as \( 5 \) units is a valid option. Step 5: Final Answer. \[ (3) \quad \mathbf{5 \text{ units}} \]