A circle is graphed in the standard (x, y) coordinate plane such that it is tangent to the… — ACT

A circle is graphed in the standard $(x, y)$ coordinate plane such that it is tangent to the $x$ -axis at $(4, 0)$ and tangent to the $y$ -axis at $(0, 4)$ . A line segment has one of its endpoints at the center of this circle and its midpoint at the point $(1, -1)$ . What is the total length of this line segment?

A
$2\sqrt{10}$
B
$\sqrt{34}$
$2\sqrt{34}$ Answer
D
$16$
E
$2\sqrt{2}$

Answer

The total length of the line segment is

2\sqrt{34}

The center of the circle is determined to be

C(4, 4)

based on the points of tangency at

(4, 0)

and

(0, 4)

. The distance between the center

C(4, 4)

and the midpoint

M(1, -1)

is calculated using the distance formula to be

\sqrt{34}

. Since the midpoint divides the segment into two equal halves, the total length of the segment is twice this distance, which is

2\sqrt{34}

Step-by-Step Solution

Determine the coordinates of the center of the circle.

The center of the circle is

C(4, 4)

Since the circle is tangent to the

x

-axis at

(4, 0)

, its center must lie on the vertical line

x = 4

. Since it is tangent to the

y

-axis at

(0, 4)

, its center must lie on the horizontal line

y = 4

. Their intersection gives the center of the circle at

C(4, 4)

Calculate the distance from the center of the circle to the midpoint of the segment.

The distance is

\sqrt{34}

Using the distance formula between the center

C(4, 4)

and the midpoint

M(1, -1)

d(C, M) = \sqrt{(1 - 4)^2 + (-1 - 4)^2} = \sqrt{(-3)^2 + (-5)^2} = \sqrt{9 + 25} = \sqrt{34}

Find the total length of the line segment.

The total length is

2\sqrt{34}

Since the midpoint divides the segment into two equal parts, the total length is twice the distance from one endpoint to the midpoint:

\text{Length} = 2 \times d(C, M) = 2\sqrt{34}

Key Concept

Using circle tangencies to find the center, and applying the distance and midpoint formulas to determine segment properties.

Estimated Time:1m 30s

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