The hypotenuse of a right triangle is 10 text{ cm}. What is the perimeter, in centimeters, of the triangle? The...

Understanding the Question

We have a right triangle with a hypotenuse of 10 cm, and we need to find its perimeter.

What We Need to Determine

The perimeter equals the sum of all three sides. Since we already know the hypotenuse (10 cm), we need to find the lengths of the two legs. Let's call them a and b.

From the Pythagorean theorem, we know that \(\mathrm{a}^2 + \mathrm{b}^2 = 10^2 = 100\). This gives us one equation with two unknowns—we need additional information to determine unique values for a and b.

Key Insight

Think of this as a constraint-counting problem: we have one constraint (\(\mathrm{a}^2 + \mathrm{b}^2 = 100\)) and two unknowns (a and b). To find a unique perimeter, we need enough additional constraints to pin down exact values for both legs.

Analyzing Statement 1

Statement 1 tells us that the area of the triangle is 25 square centimeters.

What Statement 1 Tells Us

For a right triangle, \(\mathrm{Area} = \frac{1}{2} \times \mathrm{a} \times \mathrm{b} = 25\), which means \(\mathrm{a} \times \mathrm{b} = 50\).

Now we have a system of two equations:

• \(\mathrm{a}^2 + \mathrm{b}^2 = 100\) (from the Pythagorean theorem)
• \(\mathrm{a} \times \mathrm{b} = 50\) (from the area)

With two equations and two unknowns, we can solve for unique values of a and b.

Finding the Values

From \(\mathrm{a} \times \mathrm{b} = 50\), we get \(\mathrm{b} = \frac{50}{\mathrm{a}}\). Substituting into \(\mathrm{a}^2 + \mathrm{b}^2 = 100\):

• \(\mathrm{a}^2 + \left(\frac{50}{\mathrm{a}}\right)^2 = 100\)
• \(\mathrm{a}^2 + \frac{2500}{\mathrm{a}^2} = 100\)

Multiplying through by \(\mathrm{a}^2\):

• \(\mathrm{a}^4 + 2500 = 100\mathrm{a}^2\)
• \(\mathrm{a}^4 - 100\mathrm{a}^2 + 2500 = 0\)

This is a quadratic in \(\mathrm{a}^2\). Let \(\mathrm{u} = \mathrm{a}^2\), then:

• \(\mathrm{u}^2 - 100\mathrm{u} + 2500 = 0\)
• \((\mathrm{u} - 50)^2 = 0\)
• Therefore \(\mathrm{u} = 50\), so \(\mathrm{a}^2 = 50\), which means \(\mathrm{a} = 5\sqrt{2}\)

Since \(\mathrm{a} \times \mathrm{b} = 50\) and \(\mathrm{a} = 5\sqrt{2}\), we find \(\mathrm{b} = \frac{50}{5\sqrt{2}} = 5\sqrt{2}\).

Notice that \(\mathrm{a} = \mathrm{b} = 5\sqrt{2}\), making this an isosceles right triangle!

The perimeter = \(5\sqrt{2} + 5\sqrt{2} + 10 = 10\sqrt{2} + 10 = 10(\sqrt{2} + 1)\) cm.

Conclusion

Statement 1 provides enough information to determine a unique perimeter.

[STOP - Statement 1 is Sufficient!]

This eliminates choices B, C, and E.

Analyzing Statement 2

Now we forget Statement 1 completely and analyze Statement 2 independently.

Statement 2 tells us that the two legs of the triangle are of equal length.

What Statement 2 Provides

This means \(\mathrm{a} = \mathrm{b}\), which gives us the additional constraint we need.

Key Recognition: When a right triangle has equal legs, it's an isosceles right triangle. This is a special triangle with well-known properties.

Quick Solution Using Properties

For an isosceles right triangle with hypotenuse h, each leg equals \(\frac{\mathrm{h}}{\sqrt{2}}\).

Therefore, each leg = \(\frac{10}{\sqrt{2}} = \frac{10\sqrt{2}}{2} = 5\sqrt{2}\) cm.

Let's verify this satisfies the Pythagorean theorem:

• \((5\sqrt{2})^2 + (5\sqrt{2})^2 = 50 + 50 = 100\) ✓

The perimeter = \(5\sqrt{2} + 5\sqrt{2} + 10 = 10\sqrt{2} + 10 = 10(\sqrt{2} + 1)\) cm.

Conclusion

Statement 2 provides enough information to determine a unique perimeter.

[STOP - Statement 2 is Sufficient!]

The Answer: D

Both Statement 1 alone and Statement 2 alone are sufficient to determine the perimeter.

Answer Choice D: "Each statement alone is sufficient."

Strategic Takeaway: In right triangle problems where you know one side, count your constraints versus unknowns. With the hypotenuse known, you have 1 constraint (Pythagorean theorem) and need to find 2 unknowns (the legs). Each statement that provides an independent constraint can potentially make the problem solvable.