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. None of these

Answer

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

### Step 1: Find the x-intercept (Point P)
To find the x-intercept, we set \(y = 0\) in the equation \(3x + 5y = 15\).

\[ 3x + 5(0) = 15 \implies 3x = 15 \implies x = \frac{15}{3} = 5 \]

Thus, the x-intercept is at point \(P(5, 0)\).

### Step 2: Find the y-intercept (Point Q)
To find the y-intercept, we set \(x = 0\) in the equation \(3x + 5y = 15\).

\[ 3(0) + 5y = 15 \implies 5y = 15 \implies y = \frac{15}{5} = 3 \]

Thus, the y-intercept is at point \(Q(0, 3)\).

### Step 3: Calculate the length of the hypotenuse (POQ)
The hypotenuse of the right triangle \(POQ\) can be calculated using the distance formula between points \(P(5, 0)\) and \(Q(0, 3)\).

The distance \(d\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Substituting the coordinates of points \(P\) and \(Q\):

\[ d = \sqrt{(0 - 5)^2 + (3 - 0)^2} = \sqrt{(-5)^2 + (3)^2} = \sqrt{25 + 9} = \sqrt{34} \]

Thus, the length of the hypotenuse \(POQ\) is \(\sqrt{34}\).

### Final Answer:
The hypotenuse of the right triangle \(POQ\) is of length \(\sqrt{34}\).

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