Geometry End of Course Review

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2 Area of a rectangle is equal to base x height. For a triangle Area = ½ (bh) or one half base x height. The height must be the perpendicular distance from the base to the tallest part. The area of a circle is πr 2. A 7 sided polygon is a heptagon. An 8 sided polygon is an octagon. An angle with less than 90 degrees is an acute angle. In an isosceles triangle, two sides are equal. Supplementary angles sum to 180 degrees. The simple average of a data set is the mean. A line graph may be straight, broken, curved, or double and may or may not be a function. The Fibonacci number pattern done deductively at first! is: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55. Now we will take an inductive look. (a) How do we get the next number in the series? (b) Is there a consistent pattern? Answer to (a) Add the last two numbers to get the next one. (b) There is no consistent pattern. The tangram is one of the old art forms that taught math. Kitchen floor tiled designs are good examples of tessellations. Magic squares are depicted as art in many museums and in the books of math artists and historians. The missing number in the triangle below:..\..\production Documents\GRAPHICS\Geometry\fel_geom_L2_5.jpg ( ) The inventor of the above triangle was Pascale. Geometric patterns are common in nature Some logos of automobiles use geometric shapes. The prefix geo means earth. Pascal s Triangle and Fibonacci sequence are part of the 7th grade curriculum in many school districts. One can construct planes using a ruler, protractor, compass, set squares, or T squares. An isosceles right triangle has 2 equal sides and 1 right angle. It is possible to have a scalene right triangle. (Of course it is. No sides equal.) It is not possible to have an obtuse right triangle. 2

3 Complementary angles equal 90º. Drawing and constructing don t mean the same thing. If an angle opens to the right, start counting from the right. Use a line and its mid-point to create an isosceles triangle. A circular protractor cannot be used to construct regular polygons. The prefix tells the name of the polygon. The sum of interior angles of a square is 360 degrees. The sum of interior angles of a triangle is 180 degrees. The sum of exterior angles of a polygon is 360 degrees. If the measure of the two base angles of an isosceles triangle was 80º, the measure of the third angle would be 180º - 160º = 20º. The sum of the three angles in a triangle are equal to 180º. If the sum of two angles is Xº then the third angle must be 180º - Xº. It is possible to construct a triangle with an angle of 60º and 2 sides of any equal length. Think of an isosceles triangle (2 sides equal). Any angle that is >0º and <90º is an acute angle. No triangle can have two right angles (the sum would be 180º or a straight line). No triangle can have two obtuse angles (the sum would be >180º).Therefore, a scalene triangle must have 2 acute angles. The sum of the angles in a triangle is equal to 180º. If you have two right angles, they would be equal to 180º, a straight line. Therefore you cannot have more than 1 right angle in a triangle. In any triangle ABC, the sum of the three angles is 180º. If you know Angle A and Angle B, then the sum of the two angles is Xº. The measure of the third angle, Angle C would be 180º - Xº. A B Angle A + Angle B = Xº; then angle C=180º - Xº C The angles in the trapezoid may not be congruent; therefore they would not be similar. Perimeter of any polygon = sum of all sides. Area of a triangle =1/2(bh) sup>1</sup> / ₂ (bh) Volume of a cylinder = Area of the base times height. Area is square measure Volume is cubic measure. If you divide the cubic measure by one of its components, you will have square measure. A B C D 3

4 Area of a triangle (A) is ½ the base times the height; 1/2(bh). The figure (B) is a trapezoid. The Area of a trapezoid = ½ (b 1 + b 2 ) h Perimeter of a triangle is equal to the sum of the length of the three sides. In a right triangle, the hypotenuse can be determined as follows: Hypotenuse of D: A 2 + B 2 = C 2 Assume the height of D is 12 feet; the base of D is 9 feet. First, find the length of the side opposite the right angle; the hypotenuse. From the formula = = 225. The square root of 225 = +and - 15.Now add the three sides to find the perimeter = 36. Given an equation as follows: Y + 3x = 4. Solve for Y and find the slope Solve for Y by adding -3x to each side of the equation; Y + 3x 3x = -4 3x; Y= -3x-4; slope = 3. HINT: first isolate Y on one side of the equation. An isosceles trapezoid (isosceles trapezium in British English) is a quadrilateral with a line of symmetry bisecting one pair of opposite sides, making it automatically a trapezoid. Two opposite sides are parallel, the two other sides are of equal length. Thus one pair of sides are parallel and the other pair are not. The square root of a number answers the question: What number times itself = x? Examples: 4 2 = 16 so 16 = = 100, so 100 = 10, and 144 = 12 The question is asking if the square root of the number 111 is greater than 9 and less than 10, or can be read as, is the square root of 111 between 9 and ten. Example: 144 = > 11 < 13? Is the square root of 144 between 11 and 13? In this case the answer is Yes, because 12, the square root of 144 is indeed between 11 and 13 Given a 2 + b 2 = C 2, then b = C 2 - A 2 not C 2 - a In order to verify whether or not any three sides determine a right triangle, use the Pythagorean Theorem. If side A 2 + side B 2 = side C 2, then the three side Deter mine a right triangle. Example: three sides of 5, 12, and 13. Does = 13 2? = 169. Yes, we have aright triangle. Again, three sides of 6, 13, and 14. Does = 14 2? does not equal 196. Those three sides do not determine a right triangle. 40 cm 30 cm 40 cm 4

5 In order to calculate the area of a rectangle, two dimensions are needed; the length and the width. In this problem, one dimension is missing and it can be had by finding the length of the side opposite the right angle (the hypotenuse), Pythagorean Theorem, in the given triangle by using the information given. What is the method here? If the number can be factored using a perfect square, do it; if not leave it as it is! Examples: 36 = 6 X 6; 24 = ( 4) ( 6) = 2 6 The secant is a line segment that intersects the circle at two points. The line segment originates and terminates outside the circle. The secant contains a chord. A tangent is similar to a secant except that it intersects/touches the circle at one point only. A circle that is drawn outside a polygon is circumscribed around that figure. The word circumscribed means written around. If a diameter is perpendicular to a chord, it bisects the chord and its arc and the perpendicular bisector of a chord intersects at the center of the circle If a diameter is perpendicular to a chord, it bisects the chord and its arc. 5

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