Math  /  Geometry

QuestionQuestion 54 \square 1 equilateral tilangle is inseribed I lind the length of the imadius (in emi) of the equilateral inangle.
Marke10 Hegative Mark 1025

Studdy Solution

STEP 1

1. We have a square with a circle inscribed inside it.
2. The diagonal of the square is 20220\sqrt{2} cm.
3. An equilateral triangle is inscribed inside the circle.
4. We need to find the inradius of the equilateral triangle.

STEP 2

1. Determine the side length of the square using the diagonal.
2. Find the radius of the inscribed circle (which is also the circumradius of the equilateral triangle).
3. Calculate the inradius of the equilateral triangle using its circumradius.

STEP 3

The diagonal of a square is related to its side length ss by the formula:
Diagonal=s2 \text{Diagonal} = s\sqrt{2}
Given that the diagonal is 20220\sqrt{2}, we can set up the equation:
s2=202 s\sqrt{2} = 20\sqrt{2}

STEP 4

Solve for the side length ss:
s=20 s = 20

STEP 5

The radius of the inscribed circle in the square is half the side length of the square:
Radius of circle=s2=202=10 \text{Radius of circle} = \frac{s}{2} = \frac{20}{2} = 10
This radius is also the circumradius RR of the equilateral triangle.

STEP 6

For an equilateral triangle, the relationship between the circumradius RR and the inradius rr is given by:
r=R32 r = \frac{R \sqrt{3}}{2}
Substitute R=10R = 10:
r=1032 r = \frac{10 \sqrt{3}}{2}
The length of the inradius of the equilateral triangle is:
53 \boxed{5\sqrt{3}}

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