The hypotenuse of a right triangle is 6 m more than twice the shortest side. If the third side is 2 m less than the hypotenuse, find the hypotenuse of the triangle.

Options

A. 24 m
B. 34 m
C. 26 m
D. 10 m

Answer

To solve the problem, we need to set up equations based on the information given about the sides of the right triangle.

### Step 1: Define the variables
Let:
- \( x \) = length of the shortest side
- \( h \) = length of the hypotenuse
- \( b \) = length of the third side

### Step 2: Set up the equations
From the problem statement, we have:
1. The hypotenuse is 6 m more than twice the shortest side:
\[ h = 2x + 6 \]
2. The third side is 2 m less than the hypotenuse:
\[ b = h - 2 \]

### Step 3: Use the Pythagorean theorem
In a right triangle, the relationship between the sides is given by:
\[ h^2 = x^2 + b^2 \]
Substituting the expression for \( b \) from step 2 into the Pythagorean theorem:
\[ h^2 = x^2 + (h - 2)^2 \]

### Step 4: Expand and simplify the equation
Expanding \( (h - 2)^2 \):
\[ (h - 2)^2 = h^2 - 4h + 4 \]
Now substituting this back into the Pythagorean theorem:
\[ h^2 = x^2 + h^2 - 4h + 4 \]
Subtract \( h^2 \) from both sides:
\[ 0 = x^2 - 4h + 4 \]
Rearranging gives:
\[ x^2 = 4h - 4 \]

### Step 5: Substitute \( h \) in terms of \( x \)
Now we can substitute \( h = 2x + 6 \) from step 2 into the equation:
\[ x^2 = 4(2x + 6) - 4 \]
Expanding this:
\[ x^2 = 8x + 24 - 4 \]
\[ x^2 = 8x + 20 \]

### Step 6: Rearrange the equation
Rearranging gives us a quadratic equation:
\[ x^2 - 8x - 20 = 0 \]

### Step 7: Solve the quadratic equation
We can use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1, b = -8, c = -20 \):
\[ x = \frac{-(-8) \pm \sqrt{(-8)^2 - 4 \cdot 1 \cdot (-20)}}{2 \cdot 1} \]
\[ x = \frac{8 \pm \sqrt{64 + 80}}{2} \]
\[ x = \frac{8 \pm \sqrt{144}}{2} \]
\[ x = \frac{8 \pm 12}{2} \]
Calculating the two possible values for \( x \):
1. \( x = \frac{20}{2} = 10 \)
2. \( x = \frac{-4}{2} = -2 \) (not valid since side lengths cannot be negative)

Thus, \( x = 10 \).

### Step 8: Find the hypotenuse
Now substitute \( x \) back to find \( h \):
\[ h = 2(10) + 6 = 20 + 6 = 26 \]

### Conclusion
The hypotenuse of the triangle is \( 26 \) meters.

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