https://file.aiquickdraw.com/mathbot/file/1726757597237dmxnu3uy.jpg

To prove that the points \((10, 8)\), \((2, 2)\), and \((16, 0)\) form an isosceles triangle, we need to calculate the distances between the pairs of points and verify that at least two sides have the same length.

The distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

### Step 1: Calculate the lengths of the sides of the triangle

1. **Distance between \((10, 8)\) and \((2, 2)\):**
   \[
   d_1 = \sqrt{(10 - 2)^2 + (8 - 2)^2} = \sqrt{8^2 + 6^2} = \sqrt{64 + 36} = \sqrt{100} = 10
   \]

2. **Distance between \((10, 8)\) and \((16, 0)\):**
   \[
   d_2 = \sqrt{(16 - 10)^2 + (0 - 8)^2} = \sqrt{6^2 + (-8)^2} = \sqrt{36 + 64} = \sqrt{100} = 10
   \]

3. **Distance between \((2, 2)\) and \((16, 0)\):**
   \[
   d_3 = \sqrt{(16 - 2)^2 + (0 - 2)^2} = \sqrt{14^2 + (-2)^2} = \sqrt{196 + 4} = \sqrt{200} \approx 14.14
   \]

### Step 2: Verify if the triangle is isosceles

Since two of the distances are equal (\(d_1 = d_2 = 10\)) and the third distance is different (\(d_3 \approx 14.14\)), the triangle is **isosceles**.

### Step 3: Answer the multiple-choice question

The correct formula for the distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
Thus, the correct answer is **D**.

Would you like further details or have any questions?

Here are 5 related questions to explore this concept further:
1. How do you classify triangles based on side lengths?
2. What are the properties of isosceles triangles?
3. Can you calculate the perimeter of this triangle?
4. How does the Pythagorean theorem relate to the distance formula?
5. Can you prove if the triangle is acute, right, or obtuse?

**Tip:** When solving problems involving distances between points, always use the distance formula to ensure accurate calculations.

Prove that the figure determined by the points (10,8), (2,2), and (16,0) is an isosceles triangle. Which distance formula correctly gives the distance between two points (x1,y1) and (x2,y2)?

Learn how to prove that the triangle formed by the points (10,8), (2,2), and (16,0) is isosceles using the distance formula. This problem applies coordinate geometry and basic properties of triangles, suitable for students in Grades 9-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation