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

We have three vertices: A(1, 2), B(7, 2), and C(1, 10). We need to find the radius of the incircle (the circle that fits perfectly inside the triangle, touching all three sides).

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Let's plot these points and observe something interesting:

Points A(1, 2) and B(7, 2) have the same y-coordinate (both at y = 2) so they form a horizontal line

Points A(1, 2) and C(1, 10) have the same x-coordinate (both at x = 1) so they form a vertical line

Since we have a horizontal line meeting a vertical line, we have a right triangle with the right angle at point A(1, 2).

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To find the incircle radius, we need all three side lengths. We'll use the distance formula: $d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$

This formula comes from the Pythagorean theorem - we create a right triangle between any two points and find the hypotenuse.

Side AB (horizontal):

$AB = \sqrt{(7-1)^2 + (2-2)^2} = \sqrt{6^2 + 0^2} = \sqrt{36} = 6$ units

Side AC (vertical):

$AC = \sqrt{(1-1)^2 + (10-2)^2} = \sqrt{0^2 + 8^2} = \sqrt{64} = 8$ units

Side BC (hypotenuse):

$BC = \sqrt{(1-7)^2 + (10-2)^2} = \sqrt{(-6)^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10$ units

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Let's confirm using the Pythagorean theorem: $a^2 + b^2 = c^2$

$6^2 + 8^2 = 36 + 64 = 100 = 10^2$

This confirms our triangle type and ensures our calculations are correct.

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For a right triangle: $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$

$\text{Area} = \frac{1}{2} \times 6 \times 8 = 24$ square units

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The semi-perimeter is half the perimeter:

$s = \frac{\text{perimeter}}{2} = \frac{6 + 8 + 10}{2} = \frac{24}{2} = 12$ units

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Here's the key formula: $\text{Area} = r \times s$

The incircle divides the triangle into three smaller triangles, each with the incircle center as one vertex. Each has the same height (the inradius r), and their bases are the three sides of the original triangle.

$24 = r \times 12$

$r = \frac{24}{12} = 2$ units

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For right triangles, there's an even faster formula: $r = \frac{a + b - c}{2}$

where a and b are the legs, and c is the hypotenuse.

$r = \frac{6 + 8 - 10}{2} = \frac{4}{2} = 2$ units

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The radius of the incircle is 2 units.