Lesson A Math circle level problem on area of a triangle
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<H2>A Math circle level problem on area of a triangle</H2> <H3>Problem 1</H3>Suppose a triangle with side lengths a, b, c has an in radius r=1, circumradius R=3 and a semiperimeter s=7. Find a^2 + b^2 + c^2. <B>Solution</B> This problem is above the average level of school Math problems. It is of the level of a Math circle. It requires combining several ideas and formulas. <pre> 1. Calculate the area of the triangle via inradius "r" and semi-perimeter "s" in this way: Area = r*s. (1) It gives you Area = 1*7 = 7 square units. 2. Use the Heron's formula for the area: Area = {{{sqrt(s*(s-a)*(s-b)*(s-c))}}}, which gives you 7 = {{{sqrt(7*(7-a)*(7-b)*(7-c))}}}. Square both sides to get 7^2 = 7*(7-a)*(7-b)*(7-c). Cancel the factor 7 in both sides 7 = (7-a)*(7-b)*(7-c). 7 = (49 - 7a - 7b + ab)*(7-c) = = 343 - 49a - 49b + 7ab - 49c + 7ac + 7bc - abc = = 343 - 49*(a + b + c) + 7*(ab + bc + ac) - abc. (2) 3. You are given the semi-perimeter s = 7, so you know the perimeter too: a + b + c = 7*2 = 14. (3) Substitute it into the formula (2) to get 7 = 343 - 49*14 + 7*(ab + bc + ac) - abc. (4) 4. To find abc, use the formula for the area of a triangle Area = {{{(abc)/(4*R)}}}, where R is the circumradius (5) Substituting the given and known data, it gives you 7 = {{{(abc)/(4*3)}}}, or abc = 7*4*3 = 84. (6) 5. Substitute the found value of abc into (4) to get 7 = 343 - 49*14 + 7*(ab + bc + ac) - 84. Simplify ab + bc + ac = {{{(7- 343 + 49*14 + 84)/7}}} = 62. (7) 6. Now you are in one step from getting the answer. You have a + b + c = 14. Square it: (a + b + c)^2 = 14^2 = 196 = a^2 + b^2 + c^2 + 2*(ab + ac + bc), or a^2 + b^2 + c^2 = 196 - 2*(ab + ac + bc) = 196 - 2*62 = 72. <U>Answer</U>. a^2 + b^2 + c^2 = 72. </pre> -------------- On formula (1) see the lesson - <A HREF=http://www.algebra.com/algebra/homework/Surface-area/Proof-of-the-formula-for-the-area-of-a-triangle-via-the-radius-of-the-inscribed-circle.lesson>Proof of the formula for the area of a triangle via the radius of the inscribed circle</A> in this site. On Heron's formula see the lessons - <A HREF=http://www.algebra.com/algebra/homework/Surface-area/-Proof-of-the-Heron%27s-formula-for-the-area-of-a-triangle.lesson>Proof of the Heron's formula for the area of a triangle</A>, - <A HREF=http://www.algebra.com/algebra/homework/Surface-area/One-more-proof-of-the-Heron%27s-formula-for-the-area-of-a-triangle.lesson>One more proof of the Heron's formula for the area of a triangle</A>, in this site. On formula (5) see the lesson - <A HREF=http://www.algebra.com/algebra/homework/Surface-area/Proof-of-the-formula-for-the-radius-of-the-circumscribed-circle.lesson>Proof of the formula for the radius of the circumscribed circle</A> in this site. -------------- On area of triangles see the lessons - <A HREF=http://www.algebra.com/algebra/homework/Surface-area/Formulas-for-area-of-a-triangle.lesson>Formulas for area of a triangle</A> - <A HREF=http://www.algebra.com/algebra/homework/Surface-area/-Proof-of-the-Heron%27s-formula-for-the-area-of-a-triangle.lesson>Proof of the Heron's formula for the area of a triangle</A> - <A HREF=http://www.algebra.com/algebra/homework/Surface-area/One-more-proof-of-the-Heron%27s-formula-for-the-area-of-a-triangle.lesson>One more proof of the Heron's formula for the area of a triangle</A> - <A HREF=http://www.algebra.com/algebra/homework/Surface-area/Proof-of-the-formula-for-the-area-of-a-triangle-via-the-radius-of-the-inscribed-circle.lesson>Proof of the formula for the area of a triangle via the radius of the inscribed circle</A> - <A HREF=http://www.algebra.com/algebra/homework/Surface-area/Proof-of-the-formula-for-the-radius-of-the-circumscribed-circle.lesson>Proof of the formula for the radius of the circumscribed circle</A> - <A HREF=https://www.algebra.com/algebra/homework/Surface-area/REVIEW-OF-LESSONS-ON-AREA-OF-TRIANGLES.lesson>OVERVIEW of lessons on area of triangles</A> - <A HREF=http://www.algebra.com/algebra/homework/word/geometry/Solved-problems-on-area-of-triangles.lesson>Solved problems on area of triangles</A> - <A HREF=http://www.algebra.com/algebra/homework/word/geometry/Solved-problems-on-area-of-right-angled-triangles.lesson>Solved problems on area of right-angled triangles</A> - <A HREF=http://www.algebra.com/algebra/homework/word/geometry/Solved-problems-on-area-of-regular-triangles.lesson>Solved problems on area of regular triangles</A> - <A HREF=http://www.algebra.com/algebra/homework/word/geometry/Solved-problems-on-the-radius-of-inscribed-circles-and-semicircles.lesson>Solved problems on the radius of inscribed circles and semicircles</A> - <A HREF=http://www.algebra.com/algebra/homework/word/geometry/Solved-problems-on-the-radius-of-a-circumscribed-circle.lesson>Solved problems on the radius of a circumscribed circle</A> in this site. For navigation over the lessons on <B>Area of Triangles</B> use this file/link <A HREF=https://www.algebra.com/algebra/homework/Surface-area/REVIEW-OF-LESSONS-ON-AREA-OF-TRIANGLES.lesson>OVERVIEW of lessons on area of triangles</A>. To navigate over all topics/lessons of the Online Geometry Textbook use this file/link <A HREF=https://www.algebra.com/algebra/homework/Triangles/GEOMETRY-your-online-textbook.lesson>GEOMETRY - YOUR ONLINE TEXTBOOK</A>.
