Question: The hypotenuse of a right triangle is 10 cm. What is the perimeter, in centimeters, of the triangle?

(1) The area of the triangle is 25 square centimeters.
(2) The 2 legs of the triangle are of equal length.

A) Statement (1) ALONE is sufficient, but statement (2) ALONE is not sufficient.
B) Statement (2) ALONE is sufficient, but statement (1) ALONE is not sufficient.
C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D) EACH statement ALONE is sufficient.
E) Statements (1) and (2) TOGETHER are not sufficient.

Correct Answer: D

Solution and Explanation:
Approach Solution 1:

Given that a right triangle's hypotenuse measures 10 cm, we must calculate the triangle's perimeter. We can choose to have a and b as the other two sides (or two legs) of the right triangle. The Pythagorean theorem states that because it is a right triangle, a^2 + b^2 = 10^2, or a^2 + b^2 = 100. Finding the values of a and b will allow us to calculate the triangle's perimeter, which will be equal to a + b + 10.

One Statement Only:

The triangle has a surface area of 25 square centimetres.

Given a right triangle, we have chosen to make a and b the non-hypotenuse sides. Remember that the base and height of a right triangle are actually its two non-hypotenuse sides, hence the area of this triangle is A = ab/2. From the equation a2 + b2 = 100, we may deduce that b is equal to (100 - a2). Given that A = 25, we can deduce the following by replacing A = ab/2 with b = (100 - a/2):

25 = [a√(100 - a^2)]/2

We can see that we can find a solution, even though we don't have to. And after finding the value of a, we can find the value of b because b = (100 - a2). Since we know the values of both a and b, we can thus calculate the triangle's perimeter. The question can be answered with just statement one.

Only Statement Two:

The triangle's two legs are of equal length.
Given that a = b and that a2 + b2 = 100, we can state:
a^2 + a^2 = 100

We can see that we can find a solution, even though we don't have to. And since b = an after we've found the value of a, we can also find the value of b. Since we know the values of both a and b, we can thus calculate the triangle's perimeter. The answer to the question can be found in just statement number two.

D is the correct choice.

Correct Answer: D

Approach Solution 2:

Let's assume that the right-angled triangle's height, base, and hypotenuse are all equal to a, b, and 10 cm.
We are requested to calculate the triangle's perimeter, which is equal to a + b + 10.
To determine the perimeter, you must know the value of a+b. Here, it is not required to know the specific values of a and b.
Considering that the hypotenuse is 10 cm, a^2 + b^2 = 10^2.

(1) The triangle has a 25 square-centimeter area.

Triangles have an area of (1/2) *base *height. ½ * a*b = 25
a*b = 50

(a + b)^2 = a^2 + b^2 + 2 ab = 10^2 + 2^50 = 200
A + b = \(\sqrt{200}\)

Finding the perimeter only requires knowing the value of a+b.

Therefore, assertion 1 alone suffices.

(2) The triangle's two legs are of identical length.
This assertion leads us to believe that the triangle is an isosceles triangle with a right angle. We can determine the length of the legs and the perimeter because the hypotenuse is known.
The second claim is sufficient.

D is the correct answer.

Correct Answer: D

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