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The task you have shared involves calculating the distance between the two points \((6, -5)\) and \((8, -7)\) by forming a right triangle. Let me break it down step-by-step and solve it for you.

### Step 1: Calculate the lengths of the legs (Leg 1 and Leg 2)
In the image, you are asked to graph a right triangle using the two points on the Cartesian plane. These points form the hypotenuse of the triangle.

The legs of the triangle can be calculated using the differences in the x-values and y-values of the points:
- Leg 1: Difference in x-values = \( 8 - 6 = 2 \)
- Leg 2: Difference in y-values = \( -7 - (-5) = -7 + 5 = -2 \)

### Step 2: Apply the Pythagorean Theorem
The Pythagorean Theorem states that:
\[
c^2 = a^2 + b^2
\]
Where \( c \) is the hypotenuse, and \( a \) and \( b \) are the legs.

Substituting the values:
\[
c^2 = 2^2 + (-2)^2 = 4 + 4 = 8
\]

### Step 3: Find the Hypotenuse
To find the hypotenuse \( c \), take the square root of 8:
\[
c = \sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2}
\]

So, the distance between the points (which is the hypotenuse) is \( 2\sqrt{2} \).

### Summary of Solution:
- Leg 1: 2
- Leg 2: 2
- Hypotenuse (Distance between the points): \( 2\sqrt{2} \)

Would youlike more details or have any questions about this solution? Here are five related questions you can explore:

1. How can we use the distance formula to directly calculate the distance between two points?
2. What is the significance of expressing distances in simplest radical form?
3. How would the Pythagorean theorem apply if the triangle were not right-angled?
4. Can you apply this method to any set of points on a coordinate plane?
5. What are the conditions that ensure a triangle is a right triangle?

**Tip**: When working with radicals, always check if you can simplify the square root to make the expression neater.

Graph a right triangle with the two points forming the hypotenuse. Using the sides, find the distance between the two points in simplest radical form: (6, -5) and (8, -7).

Learn how to calculate the distance between two points (6, -5) and (8, -7) using the Pythagorean theorem on a coordinate plane. This step-by-step solution shows how to graph the right triangle, find the leg lengths, and express the hypotenuse in simplest radical form. Ideal for Grades 8-10 students studying coordinate geometry and the Pythagorean theorem.

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