Please read the question carefully and select the best choice to fill in the blank

in the radius of the circle inscribed in a triangle ABC. Triangle ABC is a right- anged isosceles triangle with the hypotenuse as 62-162

It looks like the question is incomplete or possibly misphrased. You might be asking about finding the radius of the circle inscribed in a right-angled isosceles triangle with a given hypotenuse.

For a right-angled isosceles triangle, where the two legs are equal and the hypotenuse is given, the radius \(r\) of the inscribed circle can be calculated using the formula:

\[
r = \frac{a(\sqrt{2}-1)}{2}
\]
where \(a\) is the length of each leg of the triangle.

In your case, you provided the hypotenuse as \(62\), but there seems to be confusion about "162." I'll assume the hypotenuse is \(62\).

For a right-angled isosceles triangle, the relationship between the hypotenuse \(c\) and the legs \(a\) is given by:

\[
c = a\sqrt{2}
\]

So, we can find the length of the legs \(a\) by solving for \(a\):

\[
a = \frac{c}{\sqrt{2}} = \frac{62}{\sqrt{2}} = \frac{62 \times \sqrt{2}}{2} = 31\sqrt{2}
\]

Now, using this in the formula for the inscribed circle's radius:

\[
r = \frac{31\sqrt{2}(\sqrt{2}-1)}{2}
\]

Would you like further clarifications or help solving this step-by-step?

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Here are some related questions you might consider:
1. What is the relationship between the hypotenuse and legs in an isosceles right triangle?
2. How do you find the area of a right-angled isosceles triangle?
3. How can you derive the formula for the radius of an inscribed circle in different triangles?
4. What is the formula for the circumradius of a right triangle?
5. Can you find the perimeter of a right-angled isosceles triangle?

**Tip**: In an isosceles right triangle, the hypotenuse is always \(\sqrt{2}\) times longer than each of the legs.

Please read the question carefully and select the best choice to fill in the blank in the radius of the circle inscribed in a triangle ABC. Triangle ABC is a right-angled isosceles triangle with the hypotenuse as 62-162.

Learn how to calculate the radius of a circle inscribed in a right-angled isosceles triangle with a hypotenuse of 62. This solution uses the Pythagorean theorem and geometry formulas for inscribed circles. Suitable for students in Grades 9-11.

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