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A circle has center point upper O. Upper O connects to a point on the circumference labeled upper B. Upper B connects to a point outside the circle labeled upper A. A diagonal line connects upper A and upper O to form the shape of a triangle.

If BO=45 in. and BA=108 in. , then what is the length of AO to the nearest whole inch?

(1 point)
Responses

117 in.
117 in.

25 in.
25 in.

67 in.
67 in.

98 in.

1 answer

To solve for the length of \( AO \) using the information given in the question, we can apply the triangle inequality theorem.

In triangle \( AOB \):

  • \( AB \) is the length from point \( A \) to point \( B \), which is given as 108 in.
  • \( BO \) is the distance from point \( O \) (the center of the circle) to point \( B \) which is given as 45 in.
  • We want to find the length \( AO \).

According to the triangle inequality theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. Consequently, we can express this as:

  1. \( AO + BO > AB \)
  2. \( AO + AB > BO \)
  3. \( BO + AB > AO \)

Substituting the known values into the inequalities:

  1. \( AO + 45 > 108 \)

    • \( AO > 108 - 45 \)
    • \( AO > 63 \)

  2. \( AO + 108 > 45 \)

    • \( AO > 45 - 108 \)
    • This inequality is always true since \( AO \) must be a positive length.

  3. \( 45 + 108 > AO \)

    • \( 153 > AO \)

From the first and third inequalities, we have:

  • \( 63 < AO < 153 \)

To estimate \( AO \), we can look at possible options if they are close to the boundaries. The options are:

  • 117 in.
  • 25 in.
  • 67 in.
  • 98 in.

Among these, \( AO \) must be more than 63 in, and since the upper limit is below 153, let's check each option:

  • 117 in. (valid)
  • 25 in. (invalid)
  • 67 in. (valid)
  • 98 in. (valid)

By checking which of these options is likely given the specific choices, 117 inches is farthest but within the bounds and appears valid. Thus, the most appropriate length, rounded to the nearest whole inch, is 117 in., satisfying the triangle inequality.

The answer is: 117 in.