In right triangle ABC, right angled at C, M is the mid point of hypotenuse AB. C is joined to M and produced to a point B. Show that:CM=1/2AB

We are given a right triangle \( ABC \), right-angled at \( C \), with \( M \) as the midpoint of the hypotenuse \( AB \). We need to show that \( CM = \frac{1}{2}AB \).

### Step-by-Step Proof:

#### 1. Coordinates of Points:
To make the problem easier to solve, we can place the triangle \( ABC \) on a coordinate plane. Assume the following:
- \( C \) is at the origin \( (0, 0) \).
- \( A \) is at \( (a, 0) \) (on the x-axis).
- \( B \) is at \( (0, b) \) (on the y-axis).

Thus, the hypotenuse \( AB \) connects the points \( A(a, 0) \) and \( B(0, b) \).

#### 2. Midpoint \( M \) of \( AB \):
The coordinates of the midpoint \( M \) of \( AB \) can be found using the midpoint formula:
\[
M = \left( \frac{a+0}{2}, \frac{0+b}{2} \right) = \left( \frac{a}{2}, \frac{b}{2} \right)
\]

#### 3. Length of Hypotenuse \( AB \):
The length of the hypotenuse \( AB \) can be found using the distance formula:
\[
AB = \sqrt{(a - 0)^2 + (0 - b)^2} = \sqrt{a^2 + b^2}
\]

#### 4. Length of \( CM \):
Next, we calculate the length of \( CM \), the line segment from \( C(0, 0) \) to \( M\left( \frac{a}{2}, \frac{b}{2} \right) \). Using the distance formula again:
\[
CM = \sqrt{\left( \frac{a}{2} - 0 \right)^2 + \left( \frac{b}{2} - 0 \right)^2}
\]
\[
CM = \sqrt{\left( \frac{a}{2} \right)^2 + \left( \frac{b}{2} \right)^2} = \sqrt{\frac{a^2}{4} + \frac{b^2}{4}} = \sqrt{\frac{a^2 + b^2}{4}} = \frac{\sqrt{a^2 + b^2}}{2}
\]

#### 5. Conclusion:
Since \( AB = \sqrt{a^2 + b^2} \), we have:
\[
CM = \frac{1}{2}AB
\]
Thus, we have shown that \( CM = \frac{1}{2}AB \), as required.

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### Related Questions:
1. How do you find the equation of a line passing through two points in a coordinate plane?
2. What is the midpoint formula in 3D space?
3. How would you prove the Pythagorean theorem using coordinate geometry?
4. Can this method be applied to non-right triangles, and how would the proof change?
5. How do you calculate the length of a line segment in polar coordinates?

**Tip:** Always break geometry problems into simple steps like using coordinates or symmetries—it makes solving complex problems easier!

In right triangle ABC, right angled at C, M is the midpoint of hypotenuse AB. C is joined to M and produced to a point B. Show that: CM = 1/2 AB.

Explore the proof that in a right triangle ABC, with M as the midpoint of hypotenuse AB, the line segment CM equals half of AB. This solution uses coordinate geometry concepts such as the midpoint and distance formulas, and the Pythagorean theorem. Suitable for high school students in Grades 10-12.

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