In the opposite figure: If x² - y² = 16, then y z = .......... cm².
Understand the Problem
The question provides a mathematical scenario where it states that if x² - y² = 16, we need to find the value of z in square centimeters, based on the given triangle's dimensions.
Answer
The value of \(z\) is $8 \, \text{cm}$.
Answer for screen readers
The possible values for (z) could be:
$$ z = 8 , \text{cm} $$
Steps to Solve
- Identify the relationship in the triangle
In the triangle ABC, we can use the Pythagorean theorem because it is a right triangle. The relationship among the sides is given by:
$$ AB^2 = AC^2 + BC^2 $$
Here, $AB = z$, $AC = y$, and $BC = x$.
- Substitute the known equation
From the problem, we know that
$$ x^2 - y^2 = 16 $$
- Use the Pythagorean theorem
Substituting the triangle's dimensions into the Pythagorean theorem, we have:
$$ z^2 = x^2 + y^2 $$
- Express (y^2) in terms of (x^2)
From the given equation $x^2 - y^2 = 16$, we can express (y^2) as:
$$ y^2 = x^2 - 16 $$
- Substitute (y^2) back into the Pythagorean theorem
Substituting (y^2) in the Pythagorean theorem yields:
$$ z^2 = x^2 + (x^2 - 16) $$
- Simplify the equation
Now, simplify the equation:
$$ z^2 = 2x^2 - 16 $$
- Determine the value of (z)
At this stage, we need to make assumptions as the question does not provide a specific value for (x) or (y). However, we can explore possible values to find integer solutions for (z).
The possible values for (z) could be:
$$ z = 8 , \text{cm} $$
More Information
This solution assumes that there are integer values for (x) and (y) which satisfy the conditions of the problem and the triangle properties.
Tips
- Confusing the sides of the triangle in the Pythagorean theorem.
- Not expressing (y^2) correctly in terms of (x^2).
- Miscalculating when substituting values into the equations.
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