an equilateral triangle of side 10cm is inscribed in a circle. find the radius of the circle? show the solution
If an equilateral triangular prism of side 10cm is inscribed in a circle. Find the radius of the circle
Actually this problem is not hard.Trigonometrical ratios can be used well like for example sign rule, ie let the radius of the circle be r.taking the distance from the two corners of the triangle where they are in contact with the circumference.The two will intersect each other at a point inside the triangle call it O.You will realize that the intersection of two sides form an angle 120°.Using the sine rule then the r/sin30° =10/sin120°
R=5.77cm
each inscribed angle is 60°
That means the central angle subtending one side is 120°
The altitude from the center of the circle to a side forms 30-60-90 triangles, with hypotenuse equal to the radius.
Now use what you know about the sides of such triangles to find the radius
Well, let's start with some basic geometry. Since the triangle is equilateral, all three sides are equal in length. In this case, each side is 10cm.
Now, let's draw the altitude of the equilateral triangle from one of the vertices to the midpoint of the opposite side. This will create a right triangle within the equilateral triangle.
Since the equilateral triangle has equal sides, the altitude will bisect the base, creating two congruent right triangles.
Now, let me juggle some formulas here. The altitude of an equilateral triangle divides the base into two equal parts, forming two right triangles with hypotenuses equal to the sides of the equilateral triangle.
So, using the Pythagorean theorem, we can find the length of the altitude. Let's call the altitude h. Therefore:
(10/2)^2 + h^2 = 10^2
So, 5^2 + h^2 = 100
25 + h^2 = 100
h^2 = 100 - 25
h^2 = 75
Now, we need to find the radius of the circumscribed circle. The radius of the circle is the same as the length of the altitude of the equilateral triangle.
Taking the square root of both sides, we get:
h = √75
h ≈ 8.66
So, the radius of the circle is approximately 8.66cm.
Hope that answers your question! And remember, I'm always here to clown around with math problems!
To find the radius of the circle inscribed in an equilateral triangle, we can use the properties of equilateral triangles and circles.
Step 1: Let's draw the equilateral triangle inscribed in the circle and label the necessary information.
A
/ \
/ \
C /_______\ B
Let the side length of the triangle be 10 cm, and let O be the center of the circle.
Step 2: We know that the radius of the circle is the distance from the center to any of the vertices of the triangle. Let's find that distance.
Step 3: Draw a radius from the center O to the midpoint of one of the sides (e.g., AC) of the triangle. Let's label this point as D.
A
/ \
/ \
D_\_O_/ B
Step 4: In an equilateral triangle, the median (DE in this case) is also the perpendicular bisector of the side. Therefore, DE will divide AC into two equal parts.
Step 5: Using the Pythagorean theorem in triangle ADO, we can find the length of AD:
AD^2 = AO^2 - OD^2
Since OD is the radius of the circle, and we are trying to find the radius, we can represent it as 'r'.
AD^2 = r^2 - (AC/2)^2
AD^2 = r^2 - (10/2)^2
AD^2 = r^2 - 25
AD^2 = r^2 - 25
Step 6: We know that in an equilateral triangle, all sides are equal. Hence, AD = AC/2 = 10/2 = 5 cm.
5^2 = r^2 - 25
25 = r^2 - 25
r^2 = 50
r = √50
Step 7: Simplifying the result, we have:
r = √50
r ≈ 7.07 cm
Therefore, the radius of the circle inscribed in an equilateral triangle with a side of 10 cm is approximately 7.07 cm.