University:
College Entrance Preparation
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ACT Preparation (Mathematics)
Academic year:

2020
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321

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6
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HG 8.3 Notes 8.3 Medians and Altitudes of Triangles Explore Finding the Balance Point of a Triangle If a triangle were cut out of a sheet of wood or paper, the triangle could be balanced around exactly one point inside the triangle . A median of a triangle is a segment whose endpoints are a vertex of a triangle and the midpoint of the opposite side . Every triangle has three distinct medians . You can use construction tools to show that the intersection of the three medians is the balance point of the triangle . X C A ) Draw a large triangle below. Label the vertices A, B, and C. B ) Find the midpoint of the side opposite A, which is BC . You may use a compass to find points equidistant to B and C and then draw the perpendicular bisector . Or you can use paper folding or a ruler . Write the label X for the midpoint . C ) Draw a segment to connect A and X . The segment is one of the three medians of the triangle. D ) Repeat Steps B and C, this time to draw the other two medians of the triangle . Write the label Y for the midpoint of the side opposite point B, and the label Z for the midpoint of the side opposite point C. Write the label P for the intersection of the three medians. E ) Use a ruler to measure the lengths of each median and the subsegments defined by P in your triangle . Record your measurements in the table . / . ZG PX = AP = AX = Median AX : f - Z Z 5 Median BY : BY = 5„ 2 Median CZ \ CZ = £ iTl S [ BP = CP = 3> A5 PY = C PZ = F ) What pattern do you observe in the measurements ? G ) Let AX be the length of any median of a triangle from a vertex A, and let P be the intersection of the three medians . Write an equation to describe the relationship between AP and PX . A H ) Let AX be the length of any median of a triangle from a vertex A, and let P be the intersection of the three medians . Write an equation to show the relationship between AX and AP . - * HG 8, 3 Notes I ) Cut out the triangle, and then punch a very small hole through P . Stick a pencil point through the hole, and then try to spin the triangle around the pencil point. How easily does it spin ? Repeat this step with another point in the triangle, and compare the results . 1. Why is "balance point" a descriptive name for point P, the intersection of the three medians ? 2. Discussion By definition, median AX intersects AABC at points A and X . Could it intersect the triangle at a third point ? Explain why or why not . TIB Ac oJ: \ ay c h oip iAt h Sc ^ ^ Learning Target H : I can use the Centroid Theorem . £> f Af ^ Explain 1 Using the Centroid Theorem ^0 k 1( u V The intersection of the three medians of a triangle is the always inside the triangle and divides each median by the K < Y a* J f */ o rrfrz>iJ of the triangle. The centroid is Centroid Theorem The centroid theorem states that the centroid of a triangle is located 2/ 3 of the distance from each to the midpoint of the opposite side . A (AX) | AP = K U«t ) Z CP = BP = V 1 X P(centioid) Example 1 Use the Centroid Theorem to find the length. AF = 9, and CE = 7.2 B ) GE A) A G C ) GD = 7 find BG CG =| - (ce) - c C£ = 4 S C& w + &B ~ O B' * && < Of . BG -n 3 ^ inkfge IT y~ _ j Wi HG 8.3 Notes 3. To find the centroid of a triangle, how many medians of the triangle must you construct ? rLOwua cJI cxn5~ /V (M ( < / CJCA&J ' A h e(t‘ : U n t7> / 7 . Let P be the centroid of ASTU, and let SW be a median of ASTU. If SW = 18, find SP and PW . ' / f 5 w Sf f( Pu 8. In AABC, the median AD is perpendicular to BC . If AD = 21 feet, describe the position of the centroid of the triangle. a * * ‘rPwCt r i s AtA CrC\ rv 7* i {• i . f 4, C n a V * . J1 A11 2* / AV. * *? D B HG 8.3 Notes Learning Target I: I can find the intersection of medians of a triangle. Explain 2 Finding the Intersection of Medians of a Triangle When a triangle is plotted on the coordinate plane, the medians can be graphed and the location of the centroid can be identified. yj 8 < kQ (0, 8) i i . ^ - =0 tnM — Example 2 Find the coordinates of the centroid of the triangle shown on the coordinate plane. Analyze Information What does the problem ask you to find ? ; :• j/ tf 6 -- f .w« v 4 .. 1 » 2 - What information does the graph provide that will help you find the answer ? . ceS j .. 4 *w.v «%WAW *\v R (6, A ) \ X o * I 2 4 6 8 Formulate a Plan The centroid is the _ j of the medians of the triangle . Begin by of one side of the triangle . Then draw a line to calculating the . You need to draw only connect that point to a medians to find the centroid . Solve Find and plot midpoints. V N\ L Draw the medians and identify equations. SR. / I Justify and Evaluate Find the centroid. B 40 / - i lo i Cf VAW i HG 8.3 Notes Find the centroid of the triangles with the given vertices. Show your work and check your answer. 9 - P (-1, 7 ) , Q ( 9, 5 ) , R ( 4, 3 ) 10. A (-6, 0) , B ( 0, 12 ), C ( 6, 0 ) a 10 i- f T 9 a a U 7; ' * - . 5 r I- r. . l 4 ^ i b 2 • 4 r5 t t 6 7 8 9 * 4 - 10 —— 4 i "•6* -8 •7 -Gl -5 -4 ^ ^PR 35 > CerrhW £ : 2 i^-- 4 f 5-4 I *'I - T 4! l. f ~4~ f f- s; * 6j ... i j Ar« ci pt c£ 0 3 |C- %6 - 5 / D ) ^ . R, 7j m : (i S j 4) Vs T ti 4' \S ' 3; ~ia a? -2 T~ ? i <£ •3 -7 - 8! - ffi | -9 Tit f fWpt *- - 1 - 2; M . 4 J j- 2 l •1 ; ! /vtfdLpt | 1 - 10 - o' 4 \\ T ~~vp 2 .. f L t X 3 f 7 b 4 . .; t g) fhidpt PViJ pf - ¥ o * AB is X • ^ 3y £ ) V (-5,0 vS 2. • A 0 6C is AC T • C&tirovck C Learning Target J : I can identify the orthocenter of a triangle. Explain 3 Finding the Orthocenter of a Triangle Like the centroid, the orthocenter is a point that characterizes a triangle. This point of the triangle rather than the medians . involves the a An altitude of a triangle is a Q O P I C J U I I P- ¥ segment from a vertex to the line containing the opposite side . Every triangle has three altitudes . An altitude can be inside, outside, or on the triangle. & toot CX . _L yisedoc,J^ -L. In the diagram of AABC, the three altitudes are AX , BZ , and CY . Notice that two of the altitudes are outside the triangle . The length of an altitude is often called the height of a triangle . HG 8.3 Notes \- * rkeJT The V\ f & of a triangle is the intersection ( or point of concurrency ) of the lines that contain jnc the altitudes. Like the altitudes themselves, the orthocenter may be inside, outside, or on the triangle . Notice that the lines containing the altitudes are concurrent at P. The orthocenter of this triangle is P . Example : Find the orthocenter of the triangle. (j) fy 8 H“ 6 - • 4v 2 0 + 0, 2 ^ *4= 2 —— — —— 1 ' S =6 tt psrpardiculcLT i s X ~2 P (2, 6) sa Ptor OQ, i 4 1 1 6 t 0 ( 8, 0) r Li ^. PAT PQ I J Urs£, -{A NOA- _ - -X + S HNTUS- O * s* Q does | ANS: (2; 2) (j&6 C° ° ) ) ~ ( * ^ i tto orfiv»o2 PS ‘ 11. Essential Question Check- In How can you find the centroid, or balance point, of a triangle ? SfVcUrth ^ ^ . ' \ RIA < V"X !) OcSo PfaC)