The coordinates of the three vertices of a triangle are: (1, 2), (7, 2), and (1, 10). Then the radius of the incircle of the triangle is

Question

Accepted Answer

The triangle is a right-angled triangle.

![image.png](https://res.cloudinary.com/aptidude/image/upload/v1742487329/aptidude_questions/tu048stbikpkp8kygukz.png)

#### Step 1: Identify the perpendicular sides  
- The side along the $$y$$-axis is vertical: $$a = 10 - 2 = 8$$.  
- The side along the $$x$$-axis is horizontal: $$b = 7 - 1 = 6$$.

#### Step 2: Calculate the hypotenuse  
The hypotenuse $$h$$ is calculated using the Pythagorean theorem:  
$$
h = \sqrt{a^2 + b^2} = \sqrt{8^2 + 6^2} = \sqrt{64 + 36} = \sqrt{100} = 10.
$$

#### Step 3: Calculate the inradius  
The formula for the inradius of a right-angled triangle is:  
$$
	ext{Inradius} = \frac{a + b - h}{2}.
$$  
Substitute the values of $$a$$, $$b$$, and $$h$$:  
$$
	ext{Inradius} = \frac{8 + 6 - 10}{2} = \frac{4}{2} = 2.
$$

### Final Answer:  
The radius of the incircle of the triangle is **2**.