Question

select the correct answer.
triangle xyz has an altitude of h. jayden proves the area of △xyz using trigonometry. his work is shown.
the area of △xyz is \\( \frac{1}{2} y h \\) by the definition of the area of a triangle. using the sine ratio, \\( \sin (x)=\frac{h}{x} \\). also, \\( x \sin (x)=h \\) is true by the multiplication property of equality. so, by the substitution property of equality, the area of △xyz is \\( \frac{1}{2} x y \sin (x) \\).
where did jayden make his first error in the proof?
a. the sine ratio \\( \sin (x)=\frac{h}{x} \\) is incorrect. it should be \\( \sin (y)=\frac{h}{x} \\).
b. the sine ratio \\( \sin (x)=\frac{h}{x} \\) is incorrect. it should be \\( \sin (z)=\frac{h}{x} \\).
c. the area of △xyz is incorrect. it should be \\( \frac{1}{2} a h \\).
d. the area of △xyz is incorrect. it should be \\( \frac{1}{2} x h \\).

Explanation:

Step1: Recall the sine ratio formula

The sine ratio in a right - triangle is defined as \(\sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}\). In the right - triangle formed (with angle \(X\)), the side opposite to angle \(X\) is \(h\) and the hypotenuse is \(x\). But if we consider the non - right triangle area formula \(A = \frac{1}{2}ab\sin C\), when we want to express the area of \(\triangle XYZ\) using \(\sin(X)\), we should use the side opposite to the angle in the non - right triangle context.

In the right - triangle (formed by the altitude \(h\)), for angle \(Z\), \(\sin(Z)=\frac{h}{x}\) (opposite \(h\) and hypotenuse \(x\)).

Step2: Analyze each option

• Option A: \(\sin(Y)\) is not relevant in the right - triangle formed by the altitude \(h\) and side \(x\) for the purpose of deriving the non - right triangle area formula using \(\sin(X)\) or \(\sin(Z)\).
• Option B: Since in the right - triangle (with altitude \(h\)), \(\sin(Z)=\frac{h}{x}\) (opposite \(h\) and hypotenuse \(x\)), and the area of \(\triangle XYZ\) using the formula \(A=\frac{1}{2}xy\sin(Z)\) (where \(y\) is the base and \(x\) is the side adjacent to angle \(Z\)). Jayden's first error is in the sine ratio. He should have used \(\sin(Z)=\frac{h}{x}\) instead of \(\sin(X)=\frac{h}{x}\).
• Option C: The area of \(\triangle XYZ\) as \(\frac{1}{2}yh\) (where \(y\) is the base of the whole triangle) is correct by the formula \(A = \frac{1}{2}\times\text{base}\times\text{height}\).
• Option D: The area formula \(A=\frac{1}{2}yh\) (with base \(y\)) is correct, not \(\frac{1}{2}xh\) (because \(x\) is not the base of the triangle \(\triangle XYZ\)).

Answer:

B. The sine ratio \(\sin(X)=\frac{h}{x}\) is incorrect. It should be \(\sin(Z)=\frac{h}{x}\).