Question

name _____bell ___6. given that $overline{ab}$ and $overline{ad}$ are tangent to circle c.determine the value of $x$.$x =$ ___7. given $overline{ad}$ is a diameter of circle f. determine the specified arc measure and type of arc.| arc | measure | type of arc (minor, major, semicircle) || ---- | ---- | ---- || $widehat{ea}$ | | || $widehat{bca}$ | | || $widehat{bae}$ | | || $widehat{acd}$ | | |8. given that $overline{fv}$ is a diameter. determine the value of $x$.$x =$ _____

Explanation:

Problem 6

Step1: Set tangents equal

Tangents from a point to a circle are congruent, so:
$2x + 7 = 5x - 8$

Step2: Isolate x terms

Subtract $2x$ from both sides:
$7 = 3x - 8$

Step3: Solve for x

Add 8 to both sides, then divide by 3:
$15 = 3x$
$x = \frac{15}{3} = 5$

Problem 7

Step1: Calculate $\widehat{EA}$

$\overline{AD}$ is a diameter ($180^\circ$). Subtract known angles:
$m\widehat{EA} = 180^\circ - 36^\circ = 144^\circ$
This is a major arc (greater than $180^\circ$? No, correction: $144^\circ < 180^\circ$, so minor arc.

Step2: Calculate $\widehat{BCA}$

Sum angles for the arc:
$m\widehat{BCA} = 70^\circ + 180^\circ = 250^\circ$
This is a major arc (greater than $180^\circ$).

Step3: Calculate $\widehat{BAE}$

Sum all angles except $\widehat{CD}$:
$m\widehat{BAE} = 67^\circ + 180^\circ + 36^\circ = 283^\circ$
This is a major arc (greater than $180^\circ$).

Step4: Calculate $\widehat{ACD}$

$\overline{AD}$ is a semicircle plus $\widehat{CD}$:
$m\widehat{ACD} = 180^\circ + 70^\circ = 250^\circ$
This is a major arc (greater than $180^\circ$).

Problem 8

Step1: Set up diameter equation

A diameter forms a $180^\circ$ arc, so:
$6x - 8 + 140 = 180$

Step2: Simplify the equation

Combine constants:
$6x + 132 = 180$

Step3: Solve for x

Subtract 132, then divide by 6:
$6x = 48$
$x = \frac{48}{6} = 8$

Answer:

1. $x = 5$

7.

Arc	Measure	Type of Arc
$\widehat{BCA}$	$250^\circ$	Major
$\widehat{BAE}$	$283^\circ$	Major
$\widehat{ACD}$	$250^\circ$	Major

1. $x = 8$