Question

andrew and Emily are sitting on the edge of a circular sandbox at points A and E, respectively. Their mom is watching from point M. How far from Emily is their mom?​

Answer 1

The distance from their mom to Emily is;

12 ft.

Explanation:

From the given diagram of a circle and secant lines, we have by the Intersecting Secant Lines Theorem;

a × b = c × d

Where;

a = The 6 ft. line length from the point 'M' to the circle

b = the length of \overline {MA}MA = 6 ft. + 2 ft. = 8 ft.

c = The 4 ft. line length from the point 'M' to the circle

d = The length of \overline {ME}ME = 4 ft. + y

Plugging the values, gives;

6 ft. × 8 ft. = 4 ft. × (4 ft. + y)

48 ft.² = 16 ft.² + 4·y ft.

48 ft.² - 16 ft.² = 32 ft.² = 4·y ft.

y = 32 ft.²/(4 ft.) = 8 ft.

y = 8 ft.

The distance from their mom to Emily = \overline {ME}ME = 4 ft. + y

\overline {ME}ME = 4 ft. + y = 4 ft. + 8 ft. = 12 ft.

\overline {ME}ME = 12 ft.

The distance from their mom to Emily = \overline {ME}ME = 12 ft.

Answer 2

The distance from their mom to Emily is;

12 ft.

Explanation:

From the given diagram of a circle and secant lines, we have by the Intersecting Secant Lines Theorem;

a × b = c × d

Where;

a = The 6 ft. line length from the point 'M' to the circle

b = the length of
`\overline {MA}` = 6 ft. + 2 ft. = 8 ft.

c = The 4 ft. line length from the point 'M' to the circle

d = The length of
`\overline {ME}` = 4 ft. + y

Plugging the values, gives;

6 ft. × 8 ft. = 4 ft. × (4 ft. + y)

48 ft.² = 16 ft.² + 4·y ft.

48 ft.² - 16 ft.² = 32 ft.² = 4·y ft.

y = 32 ft.²/(4 ft.) = 8 ft.

y = 8 ft.

The distance from their mom to Emily =
`\overline {ME}` = 4 ft. + y

`\overline {ME}` = 4 ft. + y = 4 ft. + 8 ft. = 12 ft.

`\overline {ME}` = 12 ft.

The distance from their mom to Emily =
`\overline {ME}` = 12 ft.