Download Chapter 7 Notes - Kenston Local Schools

Chapter 7 Notes Parts of a Circle A circle is the set of all points on a flat surface that are the same distance from a fixed central point. The radius (r) is a line segment from the center to a point on the circle. The diameter (d) is a line segment drawn from one side of the circle to the other side through the center of the circle. d = 2r Congruent Triangles → Congruent Corresponding Parts If you can prove that two triangles are congruent, then you will know that all of their sides and angles are also congruent. Corresponding Parts of Congruent Triangles are Congruent (CPCTC) Regular Polygons A polygon is regular if all its sides are congruent and its angles have equal measure. An equilateral triangle and a square are each regular polygons since they are both equilateral and equiangular. See the diagrams of common regular polygons below. Reflexive Property The Reflexive Property of Equality states that the measure of any side or angle is equal to itself. A side, angle or diagonal that is shared by two shapes is always congruent. For example, in the parallelogram below, because of the Reflexive Property. Exponential Functions An exponential function has the general form y = a ∙ bx, where a is the initial value (the y­intercept) and b is the multiplier (the growth/decay). Be careful: x has to be in the exponent. For example, in the multiple representations below, the y­intercept is (0, 4) and the growth factor is 3 because the y­value is increasing by multiplying by 3. If you increase or decrease by a percentage, add (increase) or subtract (decrease) from 100% then change it to a decimal. For example: an increase of 15%: 100 + 15 = 115% = 1.15 as your multiplier a decrease of 25%: 100 ­ 25 = 75% = .75 as your multiplier Diagonals of a Rhombus A rhombus is defined as a quadrilateral with four sides of equal length. In class, you proved that the diagonals of a rhombus are perpendicular bisectors of each other. E is a midpoint of both and . Therefore, AE = CE and DE = BE. Also, m∠AEB =m∠BEC = m∠CED = m∠DEA = 90°. In addition, you proved in problem 7­82 that the diagonals bisect the angles of the rhombus. For example, in the diagram above ,m∠DAE = m∠BAE . Triangle Midsegment Theorem A midsegment of a triangle is a segment that connects the midpoints of any two sides of a triangle. Every triangle has three midsegments, as shown . A midsegment between two sides of a triangle is half the length of and parallel to the third side of the triangle. For example, in ΔABC above, is a midsegment, , and DE = AC. Coordinate Geometry Coordinate geometry is the study of geometry on a coordinate grid. As we practiced in class, you can create right triangles to solve for the lengths of each segment by using the Pythagorean Theorem. Finding the slope can help analyze the relationships between the sides of a shape. If the slopes of two sides of a shape are equal, then those sides are parallel. If the slopes of two sides of a shape are opposite reciprocals, then the sides are perpendicular (meaning they form a 90° angle). By using multiple algebraic and geometric tools, you can identify shapes. For example, further analysis of the sides and angles of ABCD above shows that AB = DC and BC = AD. Furthermore, all four angles measure 90°. These facts together indicate that ABCD must be a rectangle. Finding a Midpoint A midpoint is a point that divides a line segment into two parts of equal length. For example, M is the midpoint of AB below. Use the coordinates of the points and add half of the distance between the x­values and the y­values Example: ((1 , 3) to (5 , 8) The x­values have a difference of 4, half being 2. The y­values have a difference of 5, half being 2.5. Using the first point add the values found ( 1 + 2 , 3 + 2.5) The midpoint is (3 , 5.5) Definitions and Properties of Quadrilaterals Quadrilateral A closed four­sided polygon with straight sides. Kite A quadrilateral with two distinct pairs of consecutive congruent sides. ­ 2 pairs of congruent sides ­ one diagonal (line of symmetry) bisects the the angle ­
diagonal are perpendicular Parallelogram Rhombus A quadrilateral with two pairs of parallel sides. ­ opposite sides are parallel ­ opposite sides are congruent ­ opposite angles are congruent A quadrilateral with four sides of equal length. ­ opposite sides are parallel ­ opposite angles are congruent ­ all sides are congruent Rectangle A quadrilateral with four right angles. ­ all angles are congruent, 90° ­ opposite sides are congruent Square A quadrilateral with four sides of equal length and four right angles. ­ opposite sides are parallel ­ all sides are congruent ­ all angles are congruent, 90° Trapezoid A quadrilateral with at least one pair of parallel sides. ­ one pair or opposite sides are parallel called “bases” ­ other pair of opposite sides are called “legs” Isosceles Trapezoid A quadrilateral with at least one pair of parallel sides. ­ one pair or opposite sides are parallel called “bases” ­ base angles are congruent ­ legs are congruent