Question

in circle p, \\(\overline{pq} \cong \overline{pr}\\), \\(\overline{ab} = 7x + 2\\) and \\(\overline{cd} = 8x - 1\\). which statement is false?
answer attempt 1 out of 2
\\(\bigcirc\\) \\(dr = 51.5\\) \\(\bigcirc\\) \\(x = 5\\) \\(\bigcirc\\) \\(\overline{aq} \cong \overline{cr}\\) \\(\bigcirc\\) arc \\(ab \cong\\) arc \\(cd\\)

Explanation:

Step1: Set chords equal (congruent radii)

Since $\overline{PQ} \cong \overline{PR}$ and they are perpendicular to chords $\overline{AB}$ and $\overline{CD}$ respectively, the chords are congruent:
$7x+2 = 8x-1$

Step2: Solve for $x$

Rearrange to isolate $x$:
$2 + 1 = 8x - 7x$
$x=3$

Step3: Calculate chord length

Substitute $x=3$ into $\overline{CD}$:
$\overline{CD}=8(3)-1=23$
$\overline{DR}=\frac{1}{2}\overline{CD}=\frac{23}{2}=11.5$

Step4: Verify true statements

• $\overline{AQ}=\frac{1}{2}\overline{AB}=\frac{1}{2}(7(3)+2)=11.5$, so $\overline{AQ}\cong\overline{CR}$
• Congruent chords have congruent arcs, so Arc $AB \cong$ Arc $CD$
• $x=5$ is false (we found $x=3$), and $DR=51.5$ is false, but $x=5$ is a direct incorrect value, while $DR=51.5$ is derived from wrong $x$; the core false statement is $x=5$ and $DR=51.5$, but $x=5$ is the foundational wrong value, and $DR=51.5$ is also false. However, solving shows $x=3$, so $x=5$ is false, and $DR=11.5

eq51.5$.

Answer:

A. $\text{DR}=51.5$, B. $x=5$