Question

Angelica calculated the distance between the two points shown on the graph below.

On a coordinate plane, line A B has points (2, negative 5) and (negative 4, 5).

Step 1: Use the point C(–4, –5) to make a right triangle with the 90 degree vertex at C.

Step 2: Determine the lengths of the legs: AC = 6 and BC = 10.

Step 3: Substitute the values into the Pythagorean theorem: 10 squared = 6 squared + c squared.

Step 4: Evaluate 10 squared = 6 squared + c squared. 100 = 36 + c squared. 64 = c squared. 8 = c.

She states the length of AB is 8 units. Which best describes the accuracy of Angelica’s solution?
Angelica is correct.
Angelica made an error determining the location of the right angle. The right angle should be at (2, 5).
Angelica made an error counting the lengths of the legs of the right triangle. The lengths should be 7 and 11.
Angelica made an error substituting the values into the Pythagorean theorem. The equation should be c squared = 6 squared + 10 squared.

Answer 1

Let's analyze Angelica's solution step by step to identify any errors.

1. **Right Triangle Construction**: She constructs a right triangle with the right angle at point C(-4, -5). To find the distance AB using a right triangle, she needs to ensure that C forms a right triangle with points A(2, -5) and B(-4, 5).

2. **Calculating the Lengths of the Legs**:
- **Length AC**: The distance between A(2, -5) and C(-4, -5) is the difference in the x-coordinates:
\[
|2 - (-4)| = |2 + 4| = 6
\]
- **Length BC**: The distance between B(-4, 5) and C(-4, -5) is the difference in the y-coordinates:
\[
|5 - (-5)| = |5 + 5| = 10
\]
Thus, the lengths are correct: AC = 6 and BC = 10.

3. **Using the Pythagorean Theorem**: Angelica correctly identified that using the lengths of the legs:
\[
10^2 = 6^2 + c^2
\]
leads to:
\[
100 = 36 + c^2
\]
which simplifies correctly to \(64 = c^2\), hence \(c = 8\). This means Angelica correctly calculated the value of \(c\).

4. **Final Steps**: The hypotenuse obtained \(c\) measures the distance from point A to point B.

Now, let's find the actual distance AB using the distance formula:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
Where A(2, -5) and B(-4, 5):
\[
d = \sqrt{((-4) - 2)^2 + (5 - (-5))^2} = \sqrt{(-6)^2 + (10)^2} = \sqrt{36 + 100} = \sqrt{136} = \sqrt{4 \cdot 34} = 2\sqrt{34} \approx 11.66
\]

**Conclusion**: Angelica's calculation of the distance AB equals 8 units is incorrect because the actual distance from A to B is approximately \(11.66\) units.

The best description of the accuracy of Angelica's solution is:
**Angelica made an error determining the location of the right angle. The right angle should be at (2, 5).**

This is because she did not correctly identify the right triangle's orientation for determining the distance AB.