Where is the first error made in the proof?

[tex]\[
\begin{array}{|l|l|}
\hline
\text{Statements} & \text{Reasons} \\
\hline
\Delta XYZ \text{ with altitude } h & \text{given} \\
\hline
\sin(X) = \frac{h}{z}, \cos(X) = \frac{r}{z} & \text{definitions of sine and cosine} \\
\hline
z \sin(X) = h, z \cos(X) = r & \text{multiplication property of equality} \\
\hline
x^2 = h^2 + (y - r)^2 & \text{substitution property of equality} \\
\hline
x^2 = (z \sin(X))^2 + (y - z \cos(X))^2 & \text{substitution property of equality} \\
\hline
x^2 = z^2 \sin^2(X) + y^2 - 2yz \cos(X) + z^2 \cos^2(X) & \text{expanding the squared terms} \\
\hline
x^2 = z^2 (\sin^2(X) + \cos^2(X)) + y^2 - 2yz \cos(X) & \text{factoring} \\
\hline
\sin^2(X) = \frac{h^2}{z^2}, \cos^2(X) = \frac{r^2}{z^2} & \text{squares of sine and cosine} \\
\hline
x^2 = z^2 \left(\frac{h^2}{z^2} + \frac{r^2}{z^2}\right) + y^2 - 2yz \cos(X) & \text{substitution property of equality} \\
\hline
x^2 = z^2 \left(\frac{h^2 + r^2}{z^2}\right) + y^2 - 2yz \cos(X) & \text{addition of fractions} \\
\hline
z^2 = h^2 + r^2 & \text{Pythagorean theorem} \\
\hline
x^2 = z^2 \left(\frac{z^2}{z^2}\right) + y^2 - 2yz \cos(X) & \text{substitution property of equality} \\
\hline
x^2 = z^2 + y^2 - 2yz \cos(X) & \text{multiplicative inverse property} \\
\hline
\end{array}
\][/tex]



Answer :

The first error identified in the proof occurs in the step:
[tex]\[ z^2 = h^2 + r^2 \][/tex]

Here’s a detailed, step-by-step explanation leading up to this error and why it is incorrect:

1. Given:
Let \(\Delta XYZ\) be a triangle with an altitude \( h \) from vertex \( X \).

2. Define sine and cosine:
[tex]\[ \sin(X) = \frac{h}{z}, \quad \cos(X) = \frac{r}{z} \][/tex]
By definitions of sine and cosine in a right triangle.

3. Multiplication property of equality:
[tex]\[ z \cdot \sin(X) = h, \quad z \cdot \cos(X) = r \][/tex]
This follows directly from the definitions.

4. Substitution property of equality:
[tex]\[ x^2 = h^2 + (y - r)^2 \][/tex]

5. Substitution: Replace \( h \) and \( r \) using previous definitions:
[tex]\[ x^2 = (z \cdot \sin(X))^2 + (y - z \cdot \cos(X))^2 \][/tex]

6. Expand:
[tex]\[ x^2 = z^2 \sin^2(X) + y^2 - 2 y z \cos(X) + z^2 \cos^2(X) \][/tex]

7. Factoring out:
[tex]\[ x^2 = z^2 [\sin^2(X) + \cos^2(X)] + y^2 - 2 y z \cos(X) \][/tex]

8. Square both sides:
[tex]\[ \sin^2(X) = \frac{h^2}{z^2}, \quad \cos^2(X) = \frac{r^2}{z^2} \][/tex]

9. Substitution:
[tex]\[ x^2 = z^2 \left( \frac{h^2}{z^2} + \frac{r^2}{z^2} \right) + y^2 - 2 y z \cos(X) \][/tex]

10. Combine fractions:
[tex]\[ x^2 = z^2 \left( \frac{h^2 + r^2}{z^2} \right) + y^2 - 2 y z \cos(X) \][/tex]

11. Error – Application of Pythagorean theorem:
[tex]\[ z^2 = h^2 + r^2 \][/tex]
The Pythagorean theorem \((a^2 + b^2 = c^2)\) is incorrectly applied here. It pertains to the sides of a right triangle, but in this context \( z^2 = h^2 + r^2 \) is not accurate as \( h \) and \( r \) typically correspond to different components of the triangle, not directly related by the Pythagorean theorem outside a specific right triangle setup.

Therefore, the step [tex]\( z^2 = h^2 + r^2 \)[/tex] is the first error in the proof, as it incorrectly applies the Pythagorean theorem to parameters that do not conform to its specific conditions.