Question

the distance between city a and city b is 22 miles. the distance between city b and city c is 54 miles. the distance between city a and city c is 51 miles. what type of triangle is created by the three cities?
○ an acute triangle, because (22^2 + 54^2 > 51^2)
○ an acute triangle, because (22^2 + 51^2 > 54^2)
○ an obtuse triangle, because (22^2 + 54^2 > 51^2)
○ an obtuse triangle, because (22^2 + 51^2 > 54^2)

Explanation:

Step1: Recall Triangle Type Rule

For a triangle with side lengths \(a\), \(b\), \(c\) (where \(c\) is the longest side), we use the Pythagorean inequality:
- Acute: \(a^2 + b^2 > c^2\)
- Obtuse: \(a^2 + b^2 < c^2\)

Here, sides are \(22\), \(51\), \(54\). Longest side is \(54\), so \(c = 54\), \(a = 22\), \(b = 51\).

Step2: Calculate \(a^2 + b^2\) and \(c^2\)

\(a^2 + b^2 = 22^2 + 51^2 = 484 + 2601 = 3085\)
\(c^2 = 54^2 = 2916\)

Step3: Compare Values

Check \(a^2 + b^2\) vs \(c^2\): \(3085 > 2916\) (i.e., \(22^2 + 51^2 > 54^2\))? Wait, no—wait, original options: Wait, recheck. Wait, the options have different comparisons. Wait, let's re-express:

Wait, the sides are \(22\) (AB), \(51\) (AC), \(54\) (BC). So longest side is \(54\) (BC). So \(a = 22\), \(b = 51\), \(c = 54\).

Wait, the options: Let's check each option's inequality:

Option 1: \(22^2 + 54^2 > 51^2\) → Not relevant (should compare sum of squares of two shorter sides vs square of longest).

Option 2: \(22^2 + 51^2 > 54^2\)? Wait, \(22^2 + 51^2 = 484 + 2601 = 3085\); \(54^2 = 2916\). So \(3085 > 2916\) → \(a^2 + b^2 > c^2\) → Acute triangle. Wait, but the option says "because \(22^2 + 51^2 > 54^2\)"? Wait, no—wait the option text: "an acute triangle, because \(22^2 + 51^2 > 54^2\)"? Wait, let's re-express the options:

Wait the options are:

1. an acute triangle, because \(22^2 + 54^2 > 51^2\)
2. an acute triangle, because \(22^2 + 51^2 > 54^2\)
3. an obtuse triangle, because \(22^2 + 54^2 > 51^2\)
4. an obtuse triangle, because \(22^2 + 51^2 > 54^2\) (no, wait original options: Let's parse the image text again.

Wait the user's image:

Options:

- an acute triangle, because \(22^2 + 54^2 > 51^2\)
- an acute triangle, because \(22^2 + 51^2 > 54^2\)
- an obtuse triangle, because \(22^2 + 54^2 > 51^2\)
- an obtuse triangle, because \(22^2 + 51^2 > 54^2\) (no, wait the last option: "because \(22^2 + 51^2 > 54^2\)"? Wait, no—wait the last option: "an obtuse triangle, because \(22^2 + 51^2 > 54^2\)"? No, that can't be. Wait, maybe a typo. Wait, let's re-express:

Wait, the correct approach is: For a triangle with sides \(a \leq b \leq c\), if \(a^2 + b^2 > c^2\), acute; \(a^2 + b^2 = c^2\), right; \(a^2 + b^2 < c^2\), obtuse.

Here, \(a = 22\), \(b = 51\), \(c = 54\) (since \(22 < 51 < 54\)).

So \(a^2 + b^2 = 22^2 + 51^2 = 484 + 2601 = 3085\)
\(c^2 = 54^2 = 2916\)

Since \(3085 > 2916\) (i.e., \(a^2 + b^2 > c^2\)), the triangle is acute. Now, check which option has this inequality.

Looking at the options:

Option 2: "an acute triangle, because \(22^2 + 51^2 > 54^2\)" → This matches the calculation (since \(22^2 + 51^2 = 3085\), \(54^2 = 2916\), so \(3085 > 2916\) → \(22^2 + 51^2 > 54^2\)).

Wait, but let's check the other options:

Option 1: Compares \(22^2 + 54^2\) vs \(51^2\) → Not the sum of the two shorter sides.

So the correct option is the second one: "an acute triangle, because \(22^2 + 51^2 > 54^2\)". Wait, but wait—the original options as per the image: Let me re-express the options as per the image (from the user's input):

The options are:

1. an acute triangle, because \(22^2 + 54^2 > 51^2\)
2. an acute triangle, because \(22^2 + 51^2 > 54^2\)
3. an obtuse triangle, because \(22^2 + 54^2 > 51^2\)
4. an obtuse triangle, because \(22^2 + 51^2 > 54^2\)

Wait, no—wait the user's image text:

Looking at the vertical text (rotated 90 degrees), let's transcribe:

- an acute triangle, because \(22^2 + 54^2 > 51^2\)
- an acute triangle, because \(22^2 + 51^2 > 54^2\)
- an obtuse triangle, because \(22^2 + 54^2 > 51^2\)
- an obtuse triangle, because \(22^2 + 51^2 > 54^2\)

Wait, but mathematically, the sum of the squares of the two shorter sides (22 and 51) should be compared to the square of the longest side (54). So \(22^2 + 51^2\) vs \(54^2\). As calculated, \(22^2 + 51^2 = 3085\), \(54^2 = 2916\), so \(3085 > 2916\) → acute triangle, and the inequality is \(22^2 + 51^2 > 54^2\), which is option 2.

Wait, but let's confirm:

- For acute triangle: sum of squares of two shorter sides > square of longest side.
- Here, two shorter sides: 22, 51; longest: 54.
- \(22^2 + 51^2 = 484 + 2601 = 3085\)
- \(54^2 = 2916\)
- \(3085 > 2916\) → acute.

Thus, the correct option is the second one: "an acute triangle, because \(22^2 + 51^2 > 54^2\)".

Answer:

an acute triangle, because \(22^2 + 51^2 > 54^2\) (the second option in the list)