Question

brook considers the two diagrams shown below. in the diagrams, ( mangle cbd = mangle gfh ).
brook makes this conjecture:because the angles are equal, the proportion of area of each sector relative to the area of the given circle is also equal.
which statement describes the validity of this conjecture?
the conjecture is valid given the focus is on proportional equivalence of the two sectors.
the conjecture is not valid given the areas for the sectors are not equivalent.
the conjecture is valid given the focus is on the equivalence of the two sectors.
the conjecture is not valid given the formula for area is different for larger circles.

Explanation:

Step1: Recall the formula for the area of a sector

The area of a sector of a circle is given by \(A=\frac{\theta}{360}\times\pi r^{2}\), where \(\theta\) is the central - angle of the sector and \(r\) is the radius of the circle.

Step2: Analyze the proportion of the sector area to the circle area

The proportion of the area of a sector to the area of the circle is \(\frac{A}{\pi r^{2}}=\frac{\theta}{360}\). Since \(m\angle CBD = m\angle GFH=\theta\) (given), for the first circle with radius \(r_1\) (circle with center \(B\)) and the second circle with radius \(r_2\) (circle with center \(F\)), the proportion of the sector area to the circle area is \(\frac{\theta}{360}\) for both circles. The actual area of the sectors \(A_1=\frac{\theta}{360}\times\pi r_1^{2}\) and \(A_2=\frac{\theta}{360}\times\pi r_2^{2}\) (if \(r_1
eq r_2\), \(A_1
eq A_2\)), but the proportion \(\frac{A_1}{\pi r_1^{2}}=\frac{A_2}{\pi r_2^{2}}=\frac{\theta}{360}\)

Answer:

The conjecture is valid given the focus is on proportional equivalence of the two sectors.