VERIFYING PROPERTIES OF GEOMETRIC FIGURES. Ad is a median

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Cecily Byrd
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1 UNIT NLYTI GEOMETRY VERIFYING PROPERTIES OF GEOMETRI FIGURES Parallelogram Rhombus Quadrilateral E H D F G = D and = D EF FG GH EH I L J Right Triangle Median of a Triangle K b a c d is a median D ltitude of a Triangle E Right isector F EH is an altitude H G D is a right bisector of D Properties of quadrilaterals In order to verify properties of various quadrilaterals, you are going to need to know their characteristics. The Parallelogram 1. Opposite sides are equal in size.. Opposite angles are equal (angles a" are the same, and angles "b" are the same) The Rhombus (diamond) 1. rhombus is a 4 sided shape where all sides have the same length.. Opposite angles are equal. 3. nother interesting thing is that the diagonals (dashed lines in second figure) of a rhombus intersect each other at 90 angles. The Kite 1. kite has two pairs of equal sides. The angles are the same where the pairs meet. Diagonals (dashed lines) meet at a 90 angle, and one of the diagonal bisects (cuts equally in half) the other.

2 Summary of Properties To verify a Right Triangle 1. find all 3 slopes two must be negative reciprocals To verify a Rectangle 1. find all 4 distances opposites must be equal. find adjacent slopes must be negative reciprocals To verify a Rhombus 1. find all 4 distances must be equal. adjacent slopes must not be negative reciprocal, but opposite slopes must be equal To verify a Parallelogram 1. find all 4 slopes opposites must be equal and adjacent must not be negative reciprocal and 1. find all 4 distances opposites must be equal To verify a Square 1. find all 4 distances must be equal. find adjacent slopes must be negative reciprocals To verify a Trapezoid 1. find all 4 slopes one set of opposites slopes must be equal To find the Equation of a Right isector 1. find the midpoint of the given line segment. find the slope of the given line segment 3. find the negative reciprocal of the slope in step, which represents the slope of the right bisector Substitute the midpoint ND the slope found in step 3 in y=mx+b to find b To find the Equation of an ltitude 1. find the slope of the side opposite to the vertex. find the negative reciprocal of the slope in step 1, which represent the slope of the altitude 3. Substitute the vertex ND the slope found in the step in y=mx+b to find b Formula for a ircle centred (0,0) x y r, where r is the radius Formulas 1. distance d x x y y. midpoint 3. slope x x y y M, y y1 m x x equation of line when you have a point and the slope: a) substitute known point and slope in y=mx+b, to find out b = y intercept b) OR Substitute the known slope and the point in the slope formula y 3 x and rearrance in y=mx+b form. To find the Equation of a Median 1. find the midpoint of the side opposite to the vertex. find the slope of the median using the vertex and the midpoint 3. Substitute either the midpoint or the vertex ND the slope in y=mx+b to find b To lassify a Triangle 1. find all 3 distances all three equal equilateral two equal isosceles none equal scalene entriod of a Triangle The point of intersection of all 3 medians. To find the distance from a Point to a Line 1. Rearrange the given line in slope form. find the slope of the given line 3. find the slope of PQ - negative reciprocal of the slope in step 4. Substitute the point and the slope found in step to find b in y=mx+b and then state the equation of PQ. 5. use Elimination or Substitution between PQ and the given line to solve for point Q 6. use the length of a line formula for the line segment PQ d x x y y to find the d

3 Unit Study Notes NLYTI GEOMETRY. Distance between two points P( x1, y 1) and Q( x, y ) or Length of line segment between two points is or x y PQ x x y y. Ex 1 Find the distance between the given points. a) P 4,, Q0, 1 b) P 1,9, 3,1 Q to nearest tenth. Equation of a circle centre, (0,0) and radius = r is x y r. Ex Write an equation of the circle with (0,0) and radius of 7. Ex 3 Determine the radius of each circle. Round to the nearest tenth, if necessary. a) x y 36 b) x y 8 c) x y 81. The midpoint between two points P( x1, y 1) and Q( x, y ) is Ex 4 Find the midpoint between the given points. M PQ x1 x y1 y,. a) P( 4,), Q(0, 1) b) P( 1,9), Q (3,1) c) Finding endpoint Pg. 78#5 D. Parallel Lines have equal slopes. m1 m if and only if L1 L. Perpendicular Lines have slopes that are negative reciprocals of each other. 3 i.e.if m1, then E. Slopes m when L1 L 3 a) etween two points P( x1, y 1) and Q( x, y ) then Ex 5 Find the slope of. In other words, m1 m 1 if and only if L1 L m PQ rise y y run x x a) b) line through P( 1,9) and (3,1) y (,0) x (1, 3) Q.

4 b) Given an Equation i) y mx b, the slope is m. ii) ax by c 0, the slope is found by writing the equation in the form y mx b. Ex 6 State the slope of the following straight lines. a) 5 y x 16 b) 3x5y13 0 c) 3yx 7 d) x 4 F. Equations of lines Ex 7 Find equations of the following lines. Express you answers in the standard form of a line, x y 0 where, and I. a) m through P( 4,7) b) through R(,6), S( 5, 8) 3 c) through T( 5, ), U( 5, 4) d) perpendicular to 5x 6y 7 0 and having the same x-intercept as the line 6x 7y 4 0.

5 G. Verifying Properties of Geometric Figures Ex 8 Prove D is a parallelogram where ( 3, 3), (9,3), (1,8), and D(0,) are the vertices of the parallelogram. What is always true about the diagonals of any parallelogram? Ex 9 Points (0, 6), (0, 0) and (6, ) are vertices of a triangle. If M and N are the midpoints of and respectively, prove a) MN b) MN 1.

6 H. Distance from a point to a line. Process 1. Rearrange the given line in slope form. find the slope of the given line Q P x y 0 3. find the slope of PQ - negative reciprocal of the slope in step 4. Substitute the point and the slope found in step to find b in y=mx+b and then state the equation of PQ. 5. use Elimination or Substitution between PQ and the given line to solve for point Q use the length of a line formula for the line segment PQ d x x y y to find the d Ex 10 Find the distance, d, between the point (3,1) P and the line 3x 4y3 0. Ex. 11 Find the distance from the origin to the line x y10 0.

EQUATIONS OF ALTITUDES, MEDIANS, and PERPENDICULAR BISECTORS

EQUATIONS OF ALTITUDES, MEDIANS, and PERPENDICULAR BISECTORS Steps to Find the Median of a Triangle: -Find the midpoint of a segment using the midpoint formula. -Use the vertex and midpoint to find the