If the graph of the equation `3x + 5y = 15` cuts the coordinate axes at P and Q , then hypotenuse of right triangle POQ is of length________.

Options

A. `sqrt(17)` units
B. 5 units
C. `sqrt(34)` units
D. 4 units

Answer

To find the length of the hypotenuse of the right triangle POQ formed by the line \(3x + 5y = 15\) and the coordinate axes, we can follow these steps:

### Step 1: Find the x-intercept (Point P)
To find the x-intercept, set \(y = 0\) in the equation \(3x + 5y = 15\):
\[ 3x + 5(0) = 15 \implies 3x = 15 \implies x = 5 \]
Thus, the coordinates of point P are \((5, 0)\).

### Step 2: Find the y-intercept (Point Q)
To find the y-intercept, set \(x = 0\) in the equation \(3x + 5y = 15\):
\[ 3(0) + 5y = 15 \implies 5y = 15 \implies y = 3 \]
Thus, the coordinates of point Q are \((0, 3)\).

### Step 3: Determine the coordinates of point O
Point O is the origin, which has coordinates \((0, 0)\).

### Step 4: Calculate the length of the hypotenuse (PQ)
The hypotenuse \(PQ\) is the distance between points P and Q. We can use the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Here, let \(P = (5, 0)\) and \(Q = (0, 3)\):
\[ d = \sqrt{(0 - 5)^2 + (3 - 0)^2} = \sqrt{(-5)^2 + 3^2} = \sqrt{25 + 9} = \sqrt{34} \]

### Conclusion
The length of the hypotenuse \(PQ\) is \(\sqrt{34}\).

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