Lesson Nice geometry problem of a Math Olympiad level
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<H2>Nice geometry problem of a Math Olympiad level</H2> <H3>Problem 1</H3>The area of △ABC is 40. Points P, Q and R lie on sides AB, BC and CA respectively. If AP = 3 and PB = 5, and the area of △ABQ is equal to the area of PBQR, determine the area of △AQC. <B>Solution</B> This problem is very nice. It is nice, because the major idea of the solution is hidden - it is not on the surface and should be dug up. Due to this reason, it is a typical Math Olympiad problem. <pre> Before to move forward, make a sketch of the problem. I will assume that the sketch is in front of your eyes. Let O be the intersection point of PR and AQ. Let {{{alpha}}} be the angle ∠AOP = ∠QOR : {{{alpha}}} = ∠AOP = ∠QOR (these angles are congruent since they are vertical angles). Since the areas of triangle ABQ and quadrilateral PBQR are equal, we conclude from this fact that the areas of triangles AOP and ROQ are equal. The equation for these equal areas is {{{(1/2)*OA*OP*sin(alpha)}}} = {{{(1/2)*OQ*OR*sin(alpha)}}}. (1) After reducing, from (1) we have OA*OP = OQ*OR. (2) It leads to proportion {{{abs(OA)/abs(OR)}}} = {{{abs(OQ)/abs(OP)}}}. (3) Thus we see that triangles AOR and QOP have congruent angles AOR and QOP and proportional sides that conclude these angles. So, these triangles AOR and QOP are SIMILAR. It is the key idea of the solution. What follows, is the direct consequence of this idea. So, the triangles AOR and QOP are similar. From it, we conclude that these triangles have congruent corresponding angles. In particular, angles OAR and OQP are congruent. It implies that line AR is parallel to PQ. It is the same as to say that PQ is parallel to AC. Hence, triangle PBQ is similar to triangle ABC. From this similarity, we have {{{abs(BQ)/abs(BC)}}} = {{{abs(BP)/abs(BA)}}} = {{{5/(5+3)}}} = {{{5/8}}}. Thus we proved that under given condition, point Q divides the side BC in the same proportion {{{5/8}}} as the point P divides side AB. Hence, the area of triangle AQC is {{{3/8}}} of the area of triangle ABC, i.e. {{{(3/8)*40}}} = 3*5 = 15 square units. At this point, the problem is solved completely. </pre> My other additional lessons on miscellaneous Geometry problems in this site are - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/Find-the-rate-of-moving-of-the-tip-of-a-shadow.lesson>Find the rate of moving of the tip of a shadow</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/A-radio-transmitter-accessibility-area.lesson>A radio transmitter accessibility area</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/Miscellaneous-geometric-problems.lesson>Miscellaneous geometric problems</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/Miscellaneous-problems-on-parallelograms.lesson>Miscellaneous problems on parallelograms</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/Remarkable-properties-of-triangles-into-which-diagonals-divide-a-quadrilateral.lesson>Remarkable properties of triangles into which diagonals divide a quadrilateral</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/A-trapezoid-divided-in-four-triangles-by-its-diagonals.lesson>A trapezoid divided in four triangles by its diagonals</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/A-problem-on-heptagon.lesson>A problem on a regular heptagon</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/The-area-of-a-regular-octagon.lesson>The area of a regular octagon</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/The-fraction-of-the-area-of-a-regular-octagon.lesson>The fraction of the area of a regular octagon</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/Try-to-solve-this-nice-Geometry-problem.lesson>Try to solve these nice Geometry problems !</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/Find-the-angle-between-sides-of-folded-triangle.lesson>Find the angle between sides of folded triangle</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/A-problems-on-three-spheres.lesson>A problem on three spheres</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/A-sphere-placed-in-an-inverted-cone.lesson>A sphere placed in an inverted cone</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/07-An-upper-level-Geometry-problem-on-special-%2815-30-135%29-triangle.lesson>An upper level Geometry problem on special (15°,30°,135°)-triangle</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/A-great-Math-Olympiad-level-Geometry-problem.lesson>A great Math Olympiad level Geometry problem</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/OVERVIEW-of-my-lessons-on-additional-misc-Geometry-problems.lesson>OVERVIEW of my additional lessons on miscellaneous advanced Geometry problems</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>.
