In my book there's a picture of an equalderal triangle. This is given information. Inside the triangle all three altitudes have been drawn to find the center point of gravity. The incircle is drawn. The circumcircle is drawn.
Given that the radius of the incricle is 1, what is
24. the radius of the circumcircle
25. the circumference of each circle
26. the area of each circle
the two circles would be concentric, (have the same centre)
Draw the triangle by joining a vertex of the triangle to the centre of the incircle.
Notice that we now have a 30,60,90 triangle with the side across from the 30º equal to 1.
So the other two sides are √3 and 2, with 2 they hypotenuse.
So the inner circle has a radius of 1 (that was given)
and the outer circle has a radius of 2
You should now be able to do the rest of the question.
chord and tangent of circle
To find the radius of the circumcircle (Question 24), we can use a property of an equilateral triangle. The radius of the circumcircle is equal to one-third of the length of any side of the triangle.
Since the incircle has a radius of 1, it means that each side of the triangle is also equal to 2 times the radius of the incircle (radius = 1, so side length = 2 * 1 = 2).
Therefore, the radius of the circumcircle is one-third of the length of any side of the triangle, which is 2/3.
To find the circumference of each circle (Question 25), we can use the formula:
Circumference = 2 * π * radius
For the incircle (radius = 1), the circumference will be:
Circumference of the incircle = 2 * π * 1 = 2π
For the circumcircle (radius = 2/3), the circumference will be:
Circumference of the circumcircle = 2 * π * (2/3) = 4π/3
To find the area of each circle (Question 26), we can use the formula:
Area = π * radius^2
For the incircle (radius = 1), the area will be:
Area of the incircle = π * 1^2 = π
For the circumcircle (radius = 2/3), the area will be:
Area of the circumcircle = π * (2/3)^2 = 4π/9
To find the answers, we need to understand the properties of an equilateral triangle and the relationship between its incircle and circumcircle.
An equilateral triangle is a triangle with all three sides and angles equal. The properties of an equilateral triangle include:
1. All three sides are equal in length.
2. All three angles are equal and measure 60 degrees.
3. The altitudes of an equilateral triangle bisect the opposite sides and intersect at a common point called the centroid or center of gravity.
4. The incircle of an equilateral triangle is the circle that touches all three sides of the triangle internally. It is centered at the centroid of the triangle.
5. The circumcircle of an equilateral triangle is the circle that passes through all three vertices of the triangle. It is centered at the centroid of the triangle and intersects the triangle's sides at the midpoints.
Now let's answer each question step-by-step.
24. The radius of the circumcircle:
Since the incircle is drawn, we know that the radius of the incircle is 1. The circumcircle is centered at the centroid, which is also the center of the incircle. The radius of the circumcircle can be found by drawing a line from the centroid to any vertex of the triangle, bisecting the side it intersects. This line divides the equilateral triangle into two congruent right triangles with hypotenuse equal to the circumradius and one leg equal to half the side length (also known as the inradius). Using the Pythagorean theorem, we can find the circumradius (R) in terms of the inradius (r) as R = 2r. Therefore, the radius of the circumcircle in this case would be 2.
25. The circumference of each circle:
To find the circumference of each circle, we can use the formula C = 2πr, where C is the circumference and r is the radius.
For the incircle, since the radius is given as 1, the circumference would be 2π(1) = 2π.
For the circumcircle, since the radius is found to be 2, the circumference would be 2π(2) = 4π.
26. The area of each circle:
To find the area of each circle, we can use the formula A = πr^2, where A is the area and r is the radius.
For the incircle, the area would be π(1^2) = π.
For the circumcircle, the area would be π(2^2) = 4π.
Therefore, in this given scenario for an equilateral triangle with an incircle radius of 1, the answers would be:
24. The radius of the circumcircle is 2.
25. The circumference of each circle: 2π for the incircle, and 4π for the circumcircle.
26. The area of each circle: π for the incircle, and 4π for the circumcircle.