The line with equation 7x +5y = 70 intersects the x-axis and y-axis at A and B respectively. a) find the coordinates of the mid-point of AB and b) find the distance between A and B 

Let's solve the problem step by step.

### Given:
The equation of the line is \(7x + 5y = 70\).

### Part a) Find the coordinates of the midpoint of AB.

**Step 1: Find the coordinates of A and B.**

- **Point A (where the line intersects the x-axis):**

  On the x-axis, \(y = 0\). Substitute \(y = 0\) in the equation \(7x + 5y = 70\):

  \[
  7x + 5(0) = 70 \implies 7x = 70 \implies x = \frac{70}{7} = 10
  \]

  So, the coordinates of \(A\) are \((10, 0)\).

- **Point B (where the line intersects the y-axis):**

  On the y-axis, \(x = 0\). Substitute \(x = 0\) in the equation \(7x + 5y = 70\):

  \[
  7(0) + 5y = 70 \implies 5y = 70 \implies y = \frac{70}{5} = 14
  \]

  So, the coordinates of \(B\) are \((0, 14)\).

**Step 2: Find the midpoint of AB.**

The midpoint \(M\) of a line segment with endpoints \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

\[
M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)
\]

Substituting the coordinates of A \((10, 0)\) and B \((0, 14)\):

\[
M = \left(\frac{10 + 0}{2}, \frac{0 + 14}{2}\right) = \left(\frac{10}{2}, \frac{14}{2}\right) = (5, 7)
\]

So, the coordinates of the midpoint \(M\) are \((5, 7)\).

### Part b) Find the distance between A and B.

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

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

Substituting the coordinates of A \((10, 0)\) and B \((0, 14)\):

\[
d = \sqrt{(0 - 10)^2 + (14 - 0)^2} = \sqrt{(-10)^2 + (14)^2} = \sqrt{100 + 196} = \sqrt{296} = \sqrt{4 \times 74} = 2\sqrt{74} \approx 17.2
\]

So, the distance between \(A\) and \(B\) is \(2\sqrt{74}\) or approximately 17.2 units.

### Final Answers:
a) The coordinates of the midpoint of AB are \((5, 7)\).  
b) The distance between A and B is \(2\sqrt{74}\) or approximately 17.2 units.

Would you like any details or have any questions? 

Here are some related questions:
1. How do you derive the distance formula for two points in a plane?
2. What are the general steps to find the equation of a line given two points?
3. How can you determine the slope of a line from its equation?
4. How do you calculate the area of a triangle formed by three points?
5. How do you find the equation of a line parallel or perpendicular to a given line?

**Tip:** Always double-check your calculations, especially when working with square roots or simplifying expressions.

Learn how to find the coordinates of the midpoint and the distance between two points on a line using coordinate geometry. This problem involves the equation of a line intersecting the x-axis and y-axis, calculating the midpoint using the midpoint formula, and finding the distance between two points using the distance formula. Suitable for high school students.

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