Find the length of the hypotenuse of the right triangle whose vertices are given by the points (-2, 1), (1, 1) and (1, 2).

 

Hint: Find distance from (-2,1) to (1,2).

 

 


Find the length of the hypotenuse of the right triangle whose vertices are given by the points (-2, 1), (1, 1) and (1, 2).

 

Hint: Find distance from (-2,1) to (1,2).

 

 


10




9

To find the length of the hypotenuse of the right triangle with given vertices \((-2, 1)\), \((1, 1)\), and \((1, 2)\), we first identify the lengths of the sides of the triangle.

### Step 1: Identify the Sides of the Triangle

1. **Points A, B, and C:**
   - \( A = (-2, 1) \)
   - \( B = (1, 1) \)
   - \( C = (1, 2) \)

2. **Distances between Points:**
   - **AB:** Distance between points \(A = (-2, 1)\) and \(B = (1, 1)\).
   - **BC:** Distance between points \(B = (1, 1)\) and \(C = (1, 2)\).
   - **AC:** Distance between points \(A = (-2, 1)\) and \(C = (1, 2)\) — this will be the hypotenuse.

### Step 2: Calculate the Length of Each Side

The distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

\[
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

- **AB:**

\[
AB = \sqrt{(1 - (-2))^2 + (1 - 1)^2} = \sqrt{(1 + 2)^2 + 0^2} = \sqrt{3^2} = 3
\]

- **BC:**

\[
BC = \sqrt{(1 - 1)^2 + (2 - 1)^2} = \sqrt{0^2 + 1^2} = \sqrt{1} = 1
\]

- **AC (Hypotenuse):**

\[
AC = \sqrt{(1 - (-2))^2 + (2 - 1)^2} = \sqrt{(1 + 2)^2 + 1^2} = \sqrt{3^2 + 1^2} = \sqrt{9 + 1} = \sqrt{10}
\]

### Step 3: Conclusion

The length of the hypotenuse \(AC\) is:

\[
\boxed{\sqrt{10}}
\]

Would you like more details on any of the steps or have any questions?

### Related Questions:

1. What is the Pythagorean theorem, and how is it applied in this problem?
2. How would you find the distance between any two points in a 3D space?
3. How can we verify that the given triangle is a right triangle?
4. What are other methods to find the hypotenuse of a right triangle?
5. How do you calculate the area of this right triangle?

### Tip:
Always double-check your distance calculations with the distance formula to avoid mistakes in geometry problems.

Find the length of the hypotenuse of the right triangle whose vertices are given by the points (-2, 1), (1, 1), and (1, 2).

Learn how to calculate the hypotenuse of a right triangle with vertices at (-2, 1), (1, 1), and (1, 2). This geometry problem uses the distance formula and Pythagorean theorem, suitable for Grades 8-10.

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