Couple geometry/trigonometry questions

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In summary, the conversation revolves around finding the radius of a circle that can be drawn through the three vertices of a triangle with sides 6, 8, and 10. The first question is whether the hypotenuse of a right triangle inscribed within a circle must be the diameter, to which the answer is yes. The second question is about finding solutions to a book, to which the answer is not provided. The conversation ends with a clarification that the circle is inscribed in the triangle, not circumscribed.
  • #1
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I am reading Gelfand's Trigonometry. In one of the questions he asks: "We know from geometry that a circle may be drawn through the three vertices of any triangle. Find the radius of such a circle if the sides of the triangle are 6,8, and 10."

My first question is, I know that if the diameter of a circle is the hypotenuse of a triangle then that triangle is a right triangle. Does this imply that the hypotenuse of any right triangle inscribed within a circle must be the diameter?

If this is not the case then I'm at a loss on how to solve this problem.

Second question, I have searched around but cannot find solutions to this book is there a place to find the solutions.
 
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  • #2
Velcroe said:
I know that if the diameter of a circle is the hypotenuse of a triangle then that triangle is a right triangle. Does this imply that the hypotenuse of any right triangle inscribed within a circle must be the diameter?
Yes, it must be the diameter.
 
  • #3
Velcroe said:
I am reading Gelfand's Trigonometry. In one of the questions he asks: "We know from geometry that a circle may be drawn through the three sides of any triangle. Find the radius of such a circle if the sides of the triangle are 6,8, and 10."

My first question is, I know that if the diameter of a circle is the hypotenuse of a triangle then that triangle is a right triangle. Does this imply that the hypotenuse of any right triangle inscribed within a circle must be the diameter?

If this is not the case then I'm at a loss on how to solve this problem.

Second question, I have searched around but cannot find solutions to this book is there a place to find the solutions.
Unfortunately I cannot answer your last question. The answer to the first, however, is yes. Imagine you have the hypotenuse of a right triangle in a circle and it is not the diameter. Then for the third point to be on the circle you get either a longer side which cannot be true or an angle which cannot be right which cannot be true either.
 
  • #4
fresh_42 said:
Unfortunately I cannot answer your last question. The answer to the first, however, is yes. Imagine you have the hypotenuse of a right triangle in a circle and it is not the diameter. Then for the third point to be on the circle you get either a longer side which cannot be true or an angle which cannot be right which cannot be true either.

That makes sense, so the answer to Gelfand question quoted above would just be 5. Seems like the question is too easy which is why I asked my question in the first place. Well thank you for your response.
 
  • #5
Velcroe said:
"We know from geometry that a circle may be drawn through the three sides of any triangle.
I think the author means a circle inscribed in a triangle, instead of the opposite. He said "sides", not "corners".
Velcroe said:
I have searched around but cannot find solutions to this book is there a place to find the solutions
If assuming the circle inscribed in a triangle is correct, then will this animation be helpful?
 
  • #6
Actually my mistake in quoting the question. It actually states"... may be drawn the the three vertices of any triangle". Sorry about that don't know how I mistyped that. Fixed my original question.
 

1. What is couple geometry/trigonometry?

Couple geometry/trigonometry is a branch of mathematics that deals with the study of relationships between angles and sides of triangles, as well as the properties of shapes and figures that are formed when two or more triangles are combined.

2. What are some real-life applications of couple geometry/trigonometry?

Couple geometry/trigonometry has many applications in fields such as engineering, architecture, physics, and navigation. It is commonly used to calculate distances, heights, and angles in real-world situations, such as determining the height of a building or the distance between two points on a map.

3. What are the basic concepts of couple geometry/trigonometry?

The basic concepts of couple geometry/trigonometry include angles, sides, and ratios. Angles are measured in degrees or radians and represent the amount of rotation between two intersecting lines. Sides are the lengths of the line segments that make up a triangle. Ratios, such as sine, cosine, and tangent, relate the angles and sides of a triangle.

4. How is couple geometry/trigonometry different from regular geometry?

Couple geometry/trigonometry is an extension of regular geometry, focusing specifically on the relationships between angles and sides in triangles. It also involves using trigonometric functions to solve problems, while regular geometry typically uses algebraic equations and geometric proofs.

5. What are some common formulas used in couple geometry/trigonometry?

Some common formulas used in couple geometry/trigonometry include the Pythagorean Theorem, which relates the sides of a right triangle, and the Law of Sines and Law of Cosines, which relate the angles and sides in any triangle. Other formulas involve trigonometric functions, such as sine, cosine, and tangent, to solve for missing angles or sides in a triangle.

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