GEOMETRY

GEOMETRY

CONCEPT SUMMARY:

·          A point is a point. We typically denote points with capital letters.

·          A straight path connecting two points is called a segment, and our original two points are the endpoints of the segment. We refer to a segment by its endpoints, such as . We remove the bar to denote the length of the segment .

·          The point on a segment that is halfway between the endpoints is the midpoint of the segment. We also say that this point is equidistant from the endpoints.

·          If we start at a point, then head in one direction forever, we form a ray. Our starting point is the vertex of the ray, and we denote a ray as , where the first point is the vertex of the ray.

·          If we continue a line segment past its endpoints forever in both directions, we form a line, which we write as .

·          If this paper were continued forever in every direction, the result would be a plane. Since we can move in two general directions, such as right-left and up-down, we say the plane has two dimensions.

·          If we add a third dimension, we are in three-dimensional space.

·          The set of all points that satisfy specific conditions is called a locus.

·          The set of all points that are the same distance from a given point is a circle. The given point is the center of the circle, and the fixed distance is the radius. We often refer to a circle by its center using the symbol , so  refers to a circle centered at .

·          A line that touches a circle at a single point is tangent to the circle, while a line that hits a circle at two points is a secant line. A segment connecting two points on a circle is a chord, and a chord that passes through the center of its circle is a diameter. The portion of a circle that connects two points on the circle is an arc, which we denote with the endpoints of the arc  is the shorter arc that connects M and N.

CONCEPT SUMMARY:

·          The perimeter of a closed figure is how far you travel if you walk along its boundary all the way around it once.

·          The area of a closed figure is the number of  squares (or pieces of squares) needed to exactly cover the figure. We sometimes use brackets to denote area, so that  means the area of .

·          A rectangle has four sides and four angles, as shown. Furthermore, opposite sides of a rectangle equal each other in length.

·          The area of a rectangle with consecutive sides of length  and  is . Since a square is just a rectangle in which these lengths are the same, the area of a square equals the square of the length of one of its sides.

·          A triangle that has a right angle as one of its angles is called a right triangle. The sides adjacent to the right angle of a triangle are called the legs of the triangle.

·          The area of a right triangle is half the product of the legs of the triangle. For example, in the right triangle shown, we have

·          To find the area of a triangle, we select one side to be the base. The perpendicular segment from the vertex opposite the base to the base (extended if necessary) is the altitude. We then have

·          For example, in the triangle shown, we have

·          Area can be a very powerful problem-solving tool. One particularly useful pair of principles is:

• If two triangles share an altitude, the ratio of their areas is the ratio of the bases to which that altitude is drawn. This is particularly useful in problems in which two triangles have bases along the same line.
• If two triangles share a base, then the ratio of their areas is the ratio of the altitudes to that base.

CONCEPT SUMMARY:

·          A quadrilateral has four segments as sides, four vertices, and four angles.

·          There are two kinds of quadrilaterals: convex quadrilateral and concave quadrilateral. In school mathematics, we only deal with convex quadrilateral.

·          Example of convex quadrilateral:

·          Example of concave quadrilateral:

·          The segments connecting opposite vertices are called the diagonals of a quadrilateral.

·          The interior angles of any quadrilateral add to 360°. (Prove this!)

·          Trapezoid

§   A trapezoid is a quadrilateral with two parallel sides. The segment connecting the midpoints of the non-parallel sides is the median of the trapezoid, and the distance between the two parallel sides is the height of the trapezoid.

§   What if there are two pairs of opposite sides that are parallel like in figure below; is the quadrilateral still a trapezoid?

Good question! Unfortunately, there isn’t a good answer. Some people define a trapezoid as having exactly one pairof opposite sides, so figure above would not be a trapezoid. Other people define a trapezoid as having at least one pairof opposite sides; to these people, figure above would be considered a trapezoid. In this course, we will be careful to present proofs and definitions based on trapezoids as being valid for either of these definitions. In this section, we will focus on quadrilaterals with exactly one pair of parallel sides.

§   The median of a trapezoid is parallel to the bases of a trapezoid, and equal in length to the average of the lengths of the bases.

§   The area of a trapezoid equals the height of the trapezoid times the length of the median of the trapezoid.

§   In an isosceles trapezoid:

ü  The base angles come in two pairs of equal angles.

ü  The legs are equal.

ü  The diagonals are equal.

If any one of these items is true for a trapezoid with exactly one pair of parallel sides, then all the others must be true for that trapezoid.

·          Parallelogram

§   A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel.

§   The area of a parallelogram is the product of a side length (the base) and the distance between that side and the opposite side of the parallelogram. This distance between opposite sides is called a height of the parallelogram.

§   In parallelogram ABCD, the opposite sides are equal, the opposite angles are equal, and the diagonals bisect each other.

  Conversely, ABCD is a parallelogram if any one of the following are true:

ü   and .

ü   and .

ü  Diagonals  and  bisect each others.

Therefore, proving one of these means the other two are true.

·          Rhombus

§   A quadrilateral is a rhombus if all of its sides are equal.

§   Every rhombus is a parallelogram. Why? Therefore, everything that is true about parallelograms is true about rhombi.

§   But, every parallelogram is not a rhombus. Why? Therefore, if we prove a property of rhombi, this property is not necessarily true for all parallelograms – we’d have to prove the property for parallelograms separately.

§   The diagonals of a rhombus are perpendicular. The area of a rhombus is half the product of its diagonals (and also its base times its height).

·          Rectangles

§   A quadrilateral in which all angles are equal is a rectangle.

§   All rectangles are parallelograms. Why? So all that is true of parallelograms is true of rectangles.

§   But, every parallelograms is not rectangles. Why?

§   Let two consecutive sides of a rectangle have lengths  and . The area of the rectangle is , and the diagonals of the rectangle both have length .

§   The diagonals of a rectangle are equal to each other and equal to the square root of the sum of the squares of the length and the width of the rectangle.

·          Squares

§   A quadrilateral in which all sides are equal and all angles are equal is a square.

§   Each square is a parallelogram, a rectangle, and a rhombus so all that is true of a parallelogram, a rectangle, or a rhombus is true of a square. Why?

§   If the side length of a square is  and its diagonal is , then  , the area of the square is , or  , and the perimeter of the square is .

PROBLEM SOLVING STRATEGIES IN QUADRILATERALS:

·          Triangles, triangles, triangles. Although this chapter was about quadrilaterals, looking back you’ll see that a great many of our solutions revolved around breaking the problems into triangles on which we can use all our triangle tools.

·          Rectangles and right triangles are easier to work with than trapezoids. When stuck on a trapezoid problem, consider dropping altitudes to build rectangles and right triangles.

·          Breaking a desired angle, length, or area into parts often makes it easier to find.

·          When stuck on a problem, think “What information haven’t I used yet?” Then. Try to find some way to use that information.

·          Don’t just stare at a diagram. Set sides you seek equal to variables and find other lengths in terms of those variables. Label the diagram when you find these new lengths.

·          A picture is worth a thousand words – making a sketch can greatly help you understand a visual problem.