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Update2: See the awesome article in MathWorld on tangential quadrilaterals for more info re problem #2.
Update: See Comments section for some answers, solutions.
Part (a) of each of the following are somewhat difficult questions that can be found in some geometry textbooks. These are numerical exercises and good practice for the more difficult SAT-types of questions or for math contests. The last part of each question is an extension or generalization of the problem. Texts do not often ask students to delve beneath the surface and look for general relationships.
1. A circle of radius 4 is inscribed in a right triangle with hypotenuse 20.
(a) Find the perimeter of the triangle without using the Pythagorean Theorem. Justify your reasoning.
(b) Using the Pythagorean Theorem, show that the triangle is similar to a 3-4-5 triangle.
Note: Many students tend to guess multiples of 3-4-5 when doing these. Sometimes they get lucky but they need to prove it!(c) PROVE in general that the perimeter of a right triangle is twice the sum of its hypotenuse and the radius of its inscribed circle. Again, no Pythagorean Thm allowed.
Note: There are well-established formulas for the inradius of a triangle. Our objective here is to look at one special case.
2. A circle is inscribed in a quadrilateral which has a pair of opposite sides equal to 12 and 18. Neither pair of opposite sides of the quadrilateral is parallel.
(a) Find the perimeter of the quadrilateral. Justify your reasoning.
(b) PROVE in general that the perimeter of a quadrilateral in which a circle is inscribed equals twice the sum of either pair of opposite sides.
Note:: Not all quadrilaterals have an inscribed circle, so this is a strong condition.
Note: As always, these results need independent verification. I welcome your comments and edits!