Q. I'm confused by this. They want me to find one of angles of a triangle which has all three of its vertices at different vertices in the octagon. None of them go outside the octagon. How should I figure this out?
A. I cannot offer a specific answer, because there are many ways to inscribe a triangle in an octagon. For that matter, there are many octagons.
Are we discussing a regular octagon? If so, then it can be inscribed in a circle, and each side is a chord subtending a central angle of 45°. Suppose that each vertex of the triangle is also a vertex of the octagon. Then any angle of the triangle is also inscribed in the circle. The measure of the inscribed angle is half the corresponding central angle. If an angle intercepts one side of the octagon, then it is 22.5°. Multiply 22.5° by the number of octagon sides intercepted by the angle.
Are we discussing a regular octagon? If so, then it can be inscribed in a circle, and each side is a chord subtending a central angle of 45°. Suppose that each vertex of the triangle is also a vertex of the octagon. Then any angle of the triangle is also inscribed in the circle. The measure of the inscribed angle is half the corresponding central angle. If an angle intercepts one side of the octagon, then it is 22.5°. Multiply 22.5° by the number of octagon sides intercepted by the angle.
A tangent-tangent angle intercepts two arcs that measure 164 and 196. What is the measure of the tangent-tange?
Q. A tangent-tangent angle intercepts two arcs that measure 164 and 196. What is the measure of the tangent-tange?
Type in the correct Units, too.
Type in the correct Units, too.
A. The arcs are apparently in degrees (since they add to 360).
There's a geometric theorem that says that the measure of the angle you're asking about is equal to half the absolute value of the difference between the intercepted arcs. So
mâ θ = ½ |196° - 164°| = 16°
There's a geometric theorem that says that the measure of the angle you're asking about is equal to half the absolute value of the difference between the intercepted arcs. So
mâ θ = ½ |196° - 164°| = 16°
How would you Calculate an Arc Length Intercepted by a Rectanlge?
Q. If an arc of a circle were intercepted by a rectangle, lets say, 5 feet wide, how would you calculate the arc measure. Explain.
A. You have not given enough informayion. What is the radius? or the length of the rectangle?
Let's say the radius = r
This means that half of the diagonal of the rectangle = r.
Now, you can find the arc length of a circle using arc = rθ where θ=angle in radians.
Also, a chord length can be found using the formula chord = 2r sin(θ/2)
so given 2rsin(θ/2) = 5 and arc = rθ
Put the 2 together and you can work it out.
θ = arcsin(5/(2r))
so
arc length = r * arcsin (5 / (2r))
(5 was your rectangle width)
if rectangle width = w
arc length = r * arcsin (w / (2r))
Note:
If you don't have the radius, but you have the other dimention of the rectangle (lets call it b for breadth)
Then
w² + b² = (2r)²
2r = â(w² + b²)
so
arc length = â(w² + b²)/2 * arcsin (w /â(w² + b²) )
Let's say the radius = r
This means that half of the diagonal of the rectangle = r.
Now, you can find the arc length of a circle using arc = rθ where θ=angle in radians.
Also, a chord length can be found using the formula chord = 2r sin(θ/2)
so given 2rsin(θ/2) = 5 and arc = rθ
Put the 2 together and you can work it out.
θ = arcsin(5/(2r))
so
arc length = r * arcsin (5 / (2r))
(5 was your rectangle width)
if rectangle width = w
arc length = r * arcsin (w / (2r))
Note:
If you don't have the radius, but you have the other dimention of the rectangle (lets call it b for breadth)
Then
w² + b² = (2r)²
2r = â(w² + b²)
so
arc length = â(w² + b²)/2 * arcsin (w /â(w² + b²) )
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