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The hypotenuse of a right triangle is \(10 \text{ cm}\). What is the perimeter, in centimeters, of the triangle?
We have a right triangle with a hypotenuse of 10 cm, and we need to find its perimeter.
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.
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.
Statement 1 tells us that the area of the triangle is 25 square centimeters.
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:
With two equations and two unknowns, we can solve for unique values of a and b.
From \(\mathrm{a} \times \mathrm{b} = 50\), we get \(\mathrm{b} = \frac{50}{\mathrm{a}}\). Substituting into \(\mathrm{a}^2 + \mathrm{b}^2 = 100\):
Multiplying through by \(\mathrm{a}^2\):
This is a quadratic in \(\mathrm{a}^2\). Let \(\mathrm{u} = \mathrm{a}^2\), then:
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.
Statement 1 provides enough information to determine a unique perimeter.
[STOP - Statement 1 is Sufficient!]
This eliminates choices B, C, and E.
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.
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.
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:
The perimeter = \(5\sqrt{2} + 5\sqrt{2} + 10 = 10\sqrt{2} + 10 = 10(\sqrt{2} + 1)\) cm.
Statement 2 provides enough information to determine a unique perimeter.
[STOP - Statement 2 is Sufficient!]
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.