Question

which equation represents the value of x?
○ ( x = \frac{10 + y}{2} )
○ ( x = sqrt{100 - y^2} )
○ ( x = sqrt{y^2 + 100} )
○ ( x = 10 - y )
diagram: right triangle abc with right angle at b, ab = x, bc = y, ac = 10

Explanation:

Step1: Identify Triangle Type

The triangle \( \triangle ABC \) is a right - triangle with \( \angle B = 90^{\circ} \), hypotenuse \( AC = 10 \), one leg \( BC = y \) and the other leg \( AB=x \).

Step2: Apply Pythagorean Theorem

The Pythagorean theorem states that for a right - triangle with legs \( a \), \( b \) and hypotenuse \( c \), \( a^{2}+b^{2}=c^{2} \). In \( \triangle ABC \), let \( a = x \), \( b = y \) and \( c = 10 \). Then we have the equation \( x^{2}+y^{2}=10^{2} \), which simplifies to \( x^{2}+y^{2}=100 \).

Step3: Solve for \( x \)

We want to isolate \( x \). Subtract \( y^{2} \) from both sides of the equation \( x^{2}+y^{2}=100 \):
\( x^{2}=100 - y^{2} \)
Take the square root of both sides. Since \( x \) represents the length of a side of a triangle, \( x>0 \), so \( x=\sqrt{100 - y^{2}} \)

Answer:

\( x=\sqrt{100 - y^{2}} \) (the second option)