Geogebra

A Little Connection between Geometry and Number Theory

In this geometry problem an isosceles triangle and a right-angled triangle are inscribed in a circle. When you have solved the problem the answer may surprise you, as it has a connection to number theory. The short side of the right-angled triangle has length one and it is coincident with half of the base of the isosceles triangle which has length two. The constraint that defines the problem is that the perimeters of the two triangles are equal. What are the lengths of the other sides and what is the radius of the circle? The diagram shows that the right-angled corner is at the midpoint of the base of the isosceles triangle and that the centre of the circle is a point on the perpendicular side of the right angled triangle.

January 23, 2023 by Geogebra Geometry Mathematics 0

A Rhombus with the Least Possible Perimeter

In this geometry problem circles of radius 1 and radius 2 are tangential and fit in a rhombus such that each circle is tangential to two of its sides. The rhombus has the least possible perimeter. What is the perimeter of the rhombus and the lengths of its diagonals?

January 18, 2023 by Geogebra Geometry 1

A Fourth Problem of Circles in a Quadrilateral

As in the three preceding problems a 30-60-90° triangle is joined to a 45-90-45° triangle, long side to short side to form a composite quadrilateral. Two congruent circles are fitted that are each tangential to two sides of the quadrilateral and are tangential to each other. The diagram is shown with the 45° angle at the top and the right angle at the bottom. The circles are side by side in the diagram and fit precisely. What is the radius of the circles? Scale the diagram as convenient for your calculations.

The first of these four problems was posted here https://convexabacus.xyz/index.php/2023/01/15/a-formula-for-generating-some-geometry-problems/

January 18, 2023 by Geogebra Geometry 0

A Third Problem of Circles in a Quadrilateral

As in the preceeding problems a 30-60-90° triangle is joined to a 45-90-45° triangle, long side to short side to form a composite quadrilateral. Two congruent circles are fitted that are each tangential to two sides of the quadrilateral and are tangential to each other. The diagram is shown with the 45° angle at the top and the right angle at the bottom. The circles are above and below each other in the diagram and fit precisely. What is the radius of the circles? Scale the diagram as convenient for your calculations.

January 18, 2023 by Geogebra Geometry 0

A Second Problem of Circles in a Quadrilateral

As in the first problem a 30-60-90° triangle is joined to a 45-90-45° triangle, long side to short side to form a composite quadrilateral. In this new problem a circle is fitted lower in the diagram such that it is tangential to only the hypotenuse of the 45-90-45° triangle and to two sides of 30-60-90° triangle. What is the radius of the circle? Scale the diagram as convenient for your calculations. Which of these two problems has the bigger circle?

January 18, 2023 by Geogebra Geometry 0

A Formula for Generating some Geometry Problems

I seem to have stumbled on a formula for generating some geometry problems.

(i) Combine two familiar shapes to create a composite irregular polygon.

(ii) Fit one or two circles inside the construction.

(iii) Adjust the shapes to fit precisely.

In this problem a quadrilateral is composed of a 30-60-90° triangle abutting a 45-90-45° triangle, long side to short side. A circle is fitted that is tangential to two sides of the 45-90-45° triangle and to the short side of 30-60-90° triangle. What is the radius of the circle? Scale the diagram as convenient for your calculations. This is the first of a series of four problems.

January 15, 2023 by Geogebra Geometry 1