Question

A right triangle has sides that are consecutive even integers. The longest side is z. Which of the following equations could be used to find z?

• A. (z-4)² = z²- (z - 2)²
• B. (z-2)² = (z-4)-z²
• C. z² + 4² + 2² = 6²
• D. (z-2)² = z² - (z - 1)²
• E. (z+ 2)² + (z + 4)² = z²

Hint:

For word problems involving geometric shapes and algebraic relationships, always start by defining the variables and writing down the fundamental theorem that applies (in this case, the Pythagorean theorem). Don't forget to check if the options are just rearranged versions of your derived equation.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The problem describes a right triangle, so the relationship between its sides is governed by the Pythagorean theorem. The sides are defined as consecutive even integers.
Step 2: Key Formula or Approach:
The Pythagorean theorem states that for a right triangle with legs \(a\) and \(b\) and hypotenuse \(c\), the following relationship holds: \( a^2 + b^2 = c^2 \).
Step 3: Detailed Explanation:
We are given that the sides are consecutive even integers and the longest side is \(z\). In a right triangle, the longest side is always the hypotenuse.
If \(z\) is the longest side, and the sides are consecutive even integers, then the other two sides (the legs) must be smaller than \(z\). The even integer just before \(z\) is \(z-2\), and the even integer before that is \(z-4\).
So, the three sides of the triangle are:
- Leg 1: \( a = z - 4 \)
- Leg 2: \( b = z - 2 \)
- Hypotenuse: \( c = z \)
Now, we apply the Pythagorean theorem, \( a^2 + b^2 = c^2 \):
\[ (z-4)^2 + (z-2)^2 = z^2 \] This is the correct equation based on the problem statement. Now we must check which of the given options is equivalent to this equation.
Let's examine Option (A):
\[ (z-4)^2 = z^2 - (z-2)^2 \] To check if this is equivalent, we can add \( (z-2)^2 \) to both sides of the equation:
\[ (z-4)^2 + (z-2)^2 = z^2 - (z-2)^2 + (z-2)^2 \] \[ (z-4)^2 + (z-2)^2 = z^2 \] This is exactly the equation we derived from the Pythagorean theorem. Therefore, option (A) is correct.
Let's briefly check the other options to confirm:
(B) Incorrect algebraic manipulation.
(C) Uses constant values, not variables.
(D) Refers to consecutive integers (\(z, z-1, z-2\)), not consecutive even integers.
(E) Incorrectly assumes \(z\) is the shortest side.
Step 4: Final Answer:
The correct equation that could be used to find z is (z-4)² = z²- (z - 2)².