Algebra 1. Predicting Patterns & Examining Experiments. Unit 5: Changing on a Plane Section 4: Try Without Angles
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1 Section 4 Examines triangles in the coordinate plane, we will mention slope, but not angles (we will visit angles in Unit 6). Students will need to know the definition of collinear, isosceles, and congruent... either from prior knowledge, direct instruction, or research. Algebra 1 Predicting Patterns & Examining Experiments Unit 5: Changing on a Plane Section 4: Try Without Angles
2 What is the value of n? The corners of the three squares below are collinear. What is the value of n? (Small group discussion) Students may need to know the definition of collinear from prior knowledge, direct instruction, or research.
3 What is the value of n? The corners of the three squares below are collinear. What is the value of n? A helpful hint to get students started.
4 What is the value of n? The corners of the three squares below are collinear. What is the value of n? A : 0,4 ( ) B :( 4, 7) C :( 11,n ) Another hint, but at this point, make sure students are doing the work, don t do it for them. Struggle is important.
5 What is the value of n? The corners of the three squares below are collinear. What is the value of n? A : 0,4 ( ) B :( 4, 7) C :( 11,n ) slope AB = = 3 4 y int: 0, 4 ( ) y = 3 4 x + 4 y = 3 ( 4 11) + 4 = = 49 4 Solution - finding the equation in slope-intercept form.
6 What is the value of n? The corners of the three squares below are collinear. What is the value of n? A : 0,4 ( ) B :( 4, 7) C :( 11,n ) slope AB = = 3 4 y int: 0, 4 ( ) y = 3 4 x + 4 y = 3 ( 4 11) + 4 = = 49 4 Solution - finding the equation in slope-intercept form.
7 What is the value of n? The corners of the three squares below are collinear. What is the value of n? A : 0,4 ( ) B :( 4, 7) C :( 11,n ) slope AB = = 3 4 y int: 0, 4 ( ) y = 3 4 x + 4 y = 3 ( 4 11) + 4 = = 49 4 C : 11, 49 4 or ( 11,12.25) Solution - using the equation to find a value at x=11.
8 What is the value of n? The corners of the three squares below are collinear. What is the value of n? A : 0,4 ( ) B :( 4, 7) C :( 11,n ) slope AB = = 3 4 y int: 0, 4 ( ) C : 11, 49 4 or ( 11,12.25) y = 3 4 x + 4 y = 3 ( 4 11) + 4 = = 49 4 n = 49 4 Solution
9 (Small group discussion) Students may have a guess (especially after graphing the points), but make sure they verify it. The following solution uses distances, but just noting the slope or slope triangles would be a valid method as well. How are the triangles related? Triangles ABC and DEF have a special relationship. What is the relationship and how do you know for sure? A :( 2,4) B :( 4,5) C :( 6,1) D :( 7, 3) E :( 9,4) F :( 11,0 )
10 How are the triangles related? Triangles ABC and DEF have a special relationship. What is the relationship and how do you know for sure? A :( 2,4) B :( 4,5) C :( 6,1) D :( 7, 3) E :( 9,4) F :( 11,0 ) The triangles look like they are congruent. If the corresponding sides are the same lengths, then we will have verified our hypothesis. Solution process - Graph points and connect.
11 How are the triangles related? Triangles ABC and DEF have a special relationship. What is the relationship and how do you know for sure? A :( 2,4) B :( 4,5) C :( 6,1) D :( 7, 3) E :( 9,4) F :( 11,0 ) AC = ( 6 2) 2 + ( 1 4) 2 = = 5 DF = ( 11 7) 2 + ( 0 3) 2 = = 5 The triangles look like they are congruent. AB = ( 4 2) 2 + ( 5 4) 2 = = 5 DE = ( 9 7) 2 + ( 4 3) 2 = = 5 The corresponding sides are the same lengths, therefore the triangles are congruent. BC = 6 4 ( ) 2 + ( 1 5) 2 = = 20 EF = ( 11 9) 2 + ( 0 4) 2 = = 20 Solution process - find distances and verify hypothesis.
12 (Small group discussion) This triangle looks isosceles, but is not (by only.07), so verification is very important... we cannot always trust the way a graph looks (even one generated by a computer). Is it an isosceles triangle? The sides of the triangle below are formed by the three equations listed. Is this triangle isosceles and how do you know? 3x+2y=1 y=x-2-4x+9y=22
13 Is it an isosceles triangle? The sides of the triangle below are formed by the three equations listed. Is this triangle isosceles and how do you know? 3x+2y=1 y=x-2-4x+9y=22 3x + 2( x 2) = 1 3x + 2x 4 = 1 5x = 5 y = ( 1) 2 x = 1 y = 1 (1,-1) Solution process - Substitution to find intersection point.
14 Is it an isosceles triangle? The sides of the triangle below are formed by the three equations listed. Is this triangle isosceles and how do you know? 3x+2y=1 y=x-2-4x+9y=22 (8,6) 4x + 9( x 2) = 22 4x + 9x 18 = 22 5x = 40 y = ( 8) 2 x = 8 y = 6 (1,-1) Solution process - Substitution to find intersection point.
15 Is it an isosceles triangle? The sides of the triangle below are formed by the three equations listed. Is this triangle isosceles and how do you know? 3x+2y=1 y=x-2-4x+9y=22 4 ( 3x + 2y = 1) 3 ( 4x + 9y = 22) 12x + 8y = 4 12x + 27y = 66 35y = 70 3x + 2( 2) = 1 y = 2 3x = 3 x = 1 (-1,2) (1,-1) (8,6) Solution process - Elimination to find intersection point.
16 Is it an isosceles triangle? The sides of the triangle below are formed by the three equations listed. Is this triangle isosceles and how do you know? 3x+2y=1 y=x-2-4x+9y=22 (8,6) d = ( x 2 x ) ( y 2 y ) 2 1 (-1,2) d = ( 8 + 1) 2 + ( 6 2) 2 = = 97 d = ( 1 + 1) 2 + ( 1 2) 2 = = 13 d = ( 8 1) 2 + ( 6 + 1) 2 = = 98 (1,-1) Solution process - Distance Formula to find side-lengths.
17 Is it an isosceles triangle? The sides of the triangle below are formed by the three equations listed. Is this triangle isosceles and how do you know? 3x+2y=1 y=x-2-4x+9y=22 (8,6) d = ( x 2 x ) ( y 2 y ) 2 1 (-1,2) d = ( 8 + 1) 2 + ( 6 2) 2 = = 97 d = ( 1 + 1) 2 + ( 1 2) 2 = = 13 d = ( 8 1) 2 + ( 6 + 1) 2 = = 98 (1,-1) No, the triangle is not isosceles, because no two sides are the exact same length. Solution process - Conclusion based up results of the Distance Formula.
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