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

Question

25 in

67 in

117 in

98 in

Answer 1

To find the length of AO, you can use the triangle inequality. Here, we can use the relationship between the points:

1. The length BO = 45 in.
2. The length BA = 108 in.
3. We want to find AO.

Using the triangle inequality, we have the following:

BA + AO > BO
AO + BO > BA
BA + BO > AO

We will use the first two inequalities:

1. \( 108 + AO > 45 \)
   This simplifies to \( AO > 45 - 108 \)
   Since this will not give us any exact value, we should not rely solely on this inequality.

2. \( AO + 45 > 108 \)
   This rearranges to \( AO > 108 - 45 \)
   \( AO > 63 \)

We can see that AO must be greater than 63 inches.

Now, looking at the third inequality for more constrained bounds:

3. \( 108 + 45 > AO \)
   This rearranges to \( AO < 153 \)

From these inequalities, we know: \( 63 < AO < 153 \)

Now, let's consider the options provided:

• 25 in (too low)
• 67 in (valid)
• 117 in (valid)
• 98 in (valid)

To find the nearest whole inch that fits, let's evaluate whether it fits the triangle conditions:

1. For AO = 67 in:

   • 108 + 67 > 45 → 175 > 45 (true)
   • 67 + 45 > 108 → 112 > 108 (true)
   • 108 + 45 > 67 → 153 > 67 (true)

   ✓ All conditions satisfied.

2. For AO = 117 in:

   • 108 + 117 > 45 → 225 > 45 (true)
   • 117 + 45 > 108 → 162 > 108 (true)
   • 108 + 45 > 117 → 153 > 117 (true)

   ✓ All conditions satisfied.

3. For AO = 98 in:

   • 108 + 98 > 45 → 206 > 45 (true)
   • 98 + 45 > 108 → 143 > 108 (true)
   • 108 + 45 > 98 → 153 > 98 (true)

   ✓ All conditions satisfied.

However, the closest answer fitting all bounds given is:

• 67 in.

Thus, the length of AO to the nearest whole inch is 67 in.