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Three points determine △ABC. The distance between A and B is 43.3 feet. The distance between B and C is 12.8 feet. What is the range for the distance between A and C? The range of the distance from A to C is greater than feet and less than feet.

User Noquery
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1 Answer

6 votes

Answer:

30.5 feet ≤ AC ≤ 56.1 feet

Explanation:

To find the range of the distance between point A and point C in triangle ABC, we can use the triangle inequality theorem. According to this theorem, in a triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the third side.

In this case, we have:

The distance between A and B is 43.3 feet.

The distance between B and C is 12.8 feet.

Now, we want to find the range for the distance between A and C, which is represented by side AC. The triangle inequality for side AC is:

AC ≤ AB + BC

Substitute the given values:

AC ≤ 43.3 feet + 12.8 feet

AC ≤ 56.1 feet

So, the maximum possible distance between A and C (AC) is 56.1 feet.

Now, to find the minimum possible distance between A and C, we need to consider the possibility where the three points are collinear, and AC is the difference between AB and BC:

AC ≥ |AB - BC|

AC ≥ |43.3 feet - 12.8 feet|

AC ≥ |30.5 feet|

AC ≥ 30.5 feet

So, the minimum possible distance between A and C (AC) is 30.5 feet.

Therefore, the range for the distance between A and C is greater than or equal to 30.5 feet and less than or equal to 56.1 feet.

Mathematically;

30.5 feet ≤ AC ≤ 56.1 feet

User Donald Murray
by
8.2k points

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