Triangles, Altitudes, and Area Instructor: Natalya St. Clair

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1 Tringle, nd ltitudes erkeley Mth ircles 015 Lecture Notes Tringles, ltitudes, nd re Instructor: Ntly St. lir *Note: This M session is inspired from vriety of sources, including wesomemth, reteem Mth Zoom, Decde of Mth ircles, Vol. 1 by Zvezdelin Stnkov nd Tom Rike, nd ulgrin mthemticin nd eductor Professor Georgi Psklev. Mostly, we re very grteful to the instructors, mthemticins, nd mentors who hve inspired gret teching over the yers. 1 Introduction Geometry is the mthemtics of shpe, nd it is best understood with the help of pictures. onstruction mens to ccurtely drw picture with the help of strightedge nd compss. Sometimes you wnt to use right tringle tool, ptty pper, or protrctor. Wht cn we mke with our stright lines nd circles? In this session, we pply simple logicl rguments to the simplest of ssumptions in order to produce beutiful results. Let s begin with wrm-up problem: 1 Problem 1: Using the figure below, wht point P on the upper line should be chosen so tht the tringle formed by X, Y, nd P hs the gretest re? X Y Figure 1: Which point on the upper line results in tringle with the gretest re? 1 The question nd figure on this pge re from The Mgic of Mth by rthur enjmin. Illustrtion by Ntly St. lir. 1

2 Tringle, nd ltitudes erkeley Mth ircles 015 Lecture Notes Definition. The Distnce from Point to Line. In geometry, the shortest distnce from point to line is the perpendiculr distnce. d D Figure : D is the perpendiculr dropped from point D to line. Notice tht D line. Remrk. Remember tht D is the length of line segment D. D is number nd distnce from to D! Exercise 1: Use your strightedge to drop perpendiculr line pssing through line nd point on sheet of pper. How cn you be certin tht your line is relly the shortest distnce? Try flipping the pper upside down to drw few more perpendiculr lines. Exercise : Using the previous problem, drw the three perpendiculrs from the vertices of the tringles in the figures below. n you explin wht the height of the tringle is using this definition? Figure 3: Equilterl tringle, cute tringle, right tringle, nd obtuse tringle. Wht re the types of tringles ccording to their ngles? Using only strightedge nd right tringle tool, show tht is is possible to construct the ltitudes of the tringle. Give n lgorithm for ech construction nd prove tht it does wht is is supposed to do. onstruction 1: Given segment, find the distnces from given point to. onstruction : Drw 4 nd its three ltitudes. This figure is from The Mgic of Mth by rthur enjmin. Illustrtion by Ntly St. lir.

3 Tringle, nd ltitudes erkeley Mth ircles 015 Lecture Notes n lternte but very intuitive wy to pproch constructions 1 uses pper folding. For exmple, if you re given segment drwn on pper, you cn fold the perpendiculr segment on ptty pper., Fold firm 90 degree crese long here Figure 4: Folding the perpendiculr distnce from point to line segment. Definition. The perimeter of polygon is the sum of the lengths of its sides. We define the re of 1-by-1 squre (the unit squre) to hve re 1. When b nd h re positive integers, like in the figure below, we cn brek up the region into bh 1-by-1 squres, so its re is bh. In generl, for ny rectngle with bse b nd height h, (where b nd h re positive, but not necessrily integers) we define its re to be bh.3 h=3 b=5 Figure 5: rectngle with bse b nd height h hs perimeter b + h nd re bh. Speking of re, let s go bck to the tringle problem. Strting with the re of rectngle, it is possible to derive the re of just bout ny geometricl figure. First nd foremost, we define the re of the tringle: Definition. tringle with bse b nd height h hs re 1 bh. 3 This figure nd the one on the next pge re from The Mgic of Mth by rthur enjmin. Illustrtion by Ntly St. lir. 3

4 Tringle, nd ltitudes erkeley Mth ircles 015 Lecture Notes h h h b b b Figure 6: The re of tringle with bse b nd height h is 1 bh. This is true, regrdless of whether the tringle is right-ngled, cute, or obtuse. Problems Remrk. Nottion: We shll use [] to denote the re of tringle, [XY ZW ] to denote the re of the qudrilterl XY ZW. Formuls for res (should be memorized): tringle, rectngle, squre. 4 Shpe Perimeter re 1 Tringle + b + c bh Rectngle + b b Squre 4 = Figure 7: Vrious re nd perimeter formuls. The res of tringles (or prllelogrms) with equl bses nd equl ltitudes (heights) re equl. 1. Prove the Pythgoren Theorem using res.. If D, cn we conclude tht [] = [D]? 3. (00 M 1 #) Tringle is right tringle with s its right ngle, m = 60, nd = 10. Let P be rndomly chosen inside, nd extend P to meet t D. Wht is the probbility tht D > 5? 4. Given tht D is squre, F = G = 5, nd F = H = DE = 1, compute the re of EF GH. 4 The figures in the problems re from reteem Mth Zoom cdemy,

5 Tringle, nd ltitudes erkeley Mth ircles 015 Lecture Notes 5. In the figure D is rectngle, O = 15, O = 6, D = 8. Find the re of rectngle MNOP. 6. If the side length of n equilterl tringle is 5, wht is the re? Wht if the side length is? 7. If the side length of regulr hexgon is 5, wht is the re? Wht if the side length is? 8. The hexgons DEF nd GHJK re regulr. Find the rtio of the re of the smller hexgon to the re of the lrger. 9. The re of rectngle D is 36. E, F nd G re the midpoints of their respective sides D, D nd. H is n rbitrry point on. Find the sum of the res of the shded regions. 10. The difference in re between the lrger squre nd the smller squre is 69 nd the difference in their perimeters is 1. Find the dimensions of ech squre. 5