Points of Concurrency on a Coordinate Graph

Transcription

1 Points of Concurrency on a Coordinate Graph Name Block *Perpendicular bisectors: from the midpoint to the side opposite( ) 1. The vertices of ΔABC are A(1,6), B(5,4), C(5,-2). Find the coordinates of the Circumcenter. b) Find the midpoint of each side of the triangle P(, ) Midpoint of AB Q(, ) Midpoint of BC R(, )Midpoint of AC c) Find the slope and perpendicular slope of each side of the triangle m AB = m BC = m AC = m AB = m BC = m AC = equation though point P equation though point Q equation though point R e) Use the midpoint and the perpendicular slope to accurately draw each perpendicular bisector on the triangle. f) *Using 2 equations from above - find the intersection. (systems of equations) Find the coordinates of the Circumcenter in ΔABC by finding the point of intersection of the perpendicular bisectors 2. The vertices of ΔABC are A(-5,6), B(1, 6), and C(3, 2). Find the coordinates of the Circumcenter. b) Find the midpoint of each side of the triangle P(, ) Midpoint of AB Q(, ) Midpoint of BC R(, )Midpoint of AC c) Find the slope and perpendicular slope of each side of the triangle m AB = m BC = m AC = m AB = m BC = m AC = equation through point P equation through point Q equation through point R e) Use the midpoint and the perpendicular slope to accurately draw each perpendicular bisector on the triangle. f) *Using 2 equations from above - find the intersection. (systems of equations) Find the coordinates of the Circumcenter in ΔABC by finding the point of intersection of the perpendicular bisectors

2 3. The vertices of ΔABC are A(0, 7), B(-3, 1), and C(3, 1). Find the coordinates of the Circumcenter. b) Find the midpoint of each side of the triangle P(, ) Midpoint of AB Q(, ) Midpoint of BC R(, )Midpoint of AC c) Find the slope and perpendicular slope of each side of the triangle m AB = m BC = m AC = m AB = m BC = m AC = equation through point P equation through point Q equation through point R e) Use the midpoint and the perpendicular slope to accurately draw each perpendicular bisector on the triangle. f) *Using 2 equations from above - find the intersection. (systems of equations) Find the coordinates of the Circumcenter in ΔABC by finding the point of intersection of the perpendicular bisectors * Altitudes: From the vertex to the side opposite ( ) 4. The vertices of ΔDEF are D(5,5), E(5,-4), F(-1,-1). Find the coordinates of the Orthocenter. b) Find the slope of each side of the triangle, AND its Altitude m DE = m EF = m DF = m DE = m EF = m DF = c) Use the perpendicular slope to accurately draw each altitude on the triangle. equation through point F equation through point D equation through point E e) Find the coordinates of the Orthocenter in ΔDEF by finding the point of intersection of the altitudes

3 5. The vertices of ΔJKL are J(0, 3), K(6, -1), L(10, 3). Find the coordinates of the Orthocenter. b) Find the slope of each side of the triangle, AND its altitude m JK = m KL = m JL = m JK = m KL = m JL = c) Use the perpendicular slope to accurately draw each altitude on the triangle. d) Find the equation of each line: equation through point L equation through point J equation through point K e) Find the coordinates of the Orthocenter in ΔJKL by finding the point of intersection of the altitudes 6. The vertices of ΔABC are A(-1, -1), B(6, 6), C(6, 2). Find the coordinates of the Orthocenter. b) Find the slope of each side of the triangle, AND its altitude mab = mac = mbc = m AB = m AC = m BC = c) Use the perpendicular slope to accurately draw each altitude on the triangle. d) Find the equation of each line: equation through point C equation through point B equation through point A e) Find the coordinates of the Orthocenter in ΔPQR by finding the point of intersection of the altitudes

4 *Medians: from the vertex to a midpoint (center of gravity) 7. The vertices of ΔMNO are M(-2,5), N(6,-3), O(2,-5). Find the coordinates of the Centroid. Show all work. b) Find and label the coordinate: A(, ) Midpoint of MN B(, ) Midpoint NO C(, ) Midpoint MO c) Use the midpoint to accurately draw each median on the triangle. equation of AO equation of BM equation of CN d) *Using 2 equations from above - find the intersection. (systems of equations) Find the coordinates of the Centroid in ΔMNO by finding the point of intersection of the medians. 8. The vertices of ΔMNO are M(5, 5), N(11,-3), O(-1,1). Find the coordinates of the Centroid. Show all work. b) Find and label the coordinate: A(, ) Midpoint of MN B(, ) Midpoint NO C(, ) Midpoint MO c) Use the midpoint to accurately draw each median on the triangle. equation of AO equation of BM equation of CN d) *Using 2 equations from above - find the intersection. (systems of equations) Find the coordinates of the Centroid in ΔMNO by finding the point of intersection of the medians.

5 9. The vertices of ΔMNO are M(1, 10), N (9, 5), and O(5, 0). Find the coordinates of the Centroid. Show all work. b) Find and label the coordinate: A(, ) Midpoint of MN B(, ) Midpoint NO C(, ) Midpoint MO c) Use the midpoint to accurately draw each median on the triangle. equation of AO equation of BM equation of CN d) *Using 2 equations from above - find the intersection. (systems of equations) Find the coordinates of the Centroid in ΔMNO by finding the point of intersection of the medians. Midsegments: 10) Graph ABC: given A(-5, 1), B(-5, 7) and C( 3, 3). Find and label the midpoint of AB? (Call it point D) Find and label the midpoint of AC? (Call it point E) Is the midsegment created by these midpoints parallel to BC? Prove it! Compare the side length of the midsegment to the 3rd side length of the original triangle. PROVE IT!! Show work

6 11) Graph ABC, given A( 5, -5), B( 3,5), and C(7, - 1 ). Find and label the midpoint of AB? (Call it point D) Find and label the midpoint of AC? (Call it point E) Is the midsegment created by these midpoints parallel to BC? Prove it! Compare the side length of the midsegment to the 3rd side length of the original triangle. PROVE IT!! Show work 12) Graph HJG, given H( 5, 6), J(1,2), and G(-7, - 2 ). Find and label the midpoint of HG? (Call it point K) Find and label the midpoint of HJ? (Call it point L) Is the midsegment created by these midpoints parallel to GJ? Prove it! Compare the side length of the midsegment to the 3rd side length of the original triangle. PROVE IT!! Show work