Tactics 1 to 4: Dealing with Diagrams

1. Tactics 1 to 4: Dealing with Diagrams Tactic 1: Draw a Diagram

2. DraW A Diagram • Solving Geometry problems is often easier with a diagram to look at. • Many other types of S.A.T. Problems can be aided by drawing a diagram or chart. • For this tactic, you will need a piece of paper with a straight edge. This will symbolize your answer sheet on the actual S.A.T.

3. Example 1.1 What is the area of a rectangle whose length is twice its width and whose perimeter is equal to that of a square whose area is 1? 1 Area = x2 2w w x w 1 1 1 = x2 2w 1 = x 1 x Perimeter = 4 Perimeter = 4 = 6w 4 /6 = w 2(2/3) = L 2 /3 = w 4 /3 = L 8/9 A = LW = 2/3 x 4/3

4. Example 1.2 The diagonal of square A is equal to the perimeter of square B. The area of square A is how many times the area of square B? A) 2 B) C) 4 D) E) 8 Will more than 2 B’s fit? B Will more than 4 B’s fit? 1 1

5. Example 1.2 The diagonal of square A is equal to the perimeter of square B. The area of square A is how many times the area of square B? A) 2 B) C) 4 D) E) 8 B x 1 x2 + x2 = 42 2x2 = 16 1 4 x2 = 8 Area B = 1 Area A = 8 Area A = x2 x

6. Example 1.3 • In 1995, Diana read 10 English books and 7 French books. In 1996, she read twice as many French books as English books. If 60% of the books that she read during the 2 years were French, how many English and French books did she read in 1996? .60 (2 year total) = French .60 (10+7+x+2x) = 7+2x 10 7 .60 (17 + 3x) = 7+2x 10.2 +1.8x = 7+2x x 2x 16 32 3.2 = 0.2x 16 = x 48

7. Tactics 1 to 4: Dealing with Diagrams Tactic 2: If a Diagram is Drawn to Scale, Trust It, then Trust Your Eyes

8. DIAGRAMS On the S.A.T. every attempt possible has been made to draw figures as accurately as possible. For the most part, you can trust your eyes. For example, parallel lines look parallel, right angles look 90 degrees, and distances are what they seem. If a diagram is not drawn to scale, you will be warned: “Note: Figure not drawn to scale.”

9. A F G B H C D I J E Example 2.1 In the figure at the right, EF, not shown, is a diagonal of rectangle AFJE and a diameter of the circle. D is the midpoint of AE, C is the midpoint of AD, and B is the midpoint of AC. If AE is 8 and the radius of the circle is 5, what is the area of rectangle BGHC? A)4 B) 6 C) 8 D) 12 E) 24 Area = 1 x BG 1 2 Does BG “look” bigger or smaller than AE? AE = 8 Does BG “look” bigger or smaller than DE? DE = 4 4

10. A w° P z° O x° y° B IMPORTANT: W and Y actually are exactly 90 degrees. A radius and a tangent line always form a right angle. Example 2.2 In the diagram , rays PA and PB are tangent to circle O. Which of the following is equal to z? A) x B) 180 – x C) w + x + y D) (w + x + y) /2 E) (w + x + y) /3 (90+90+90) = 270 (270)/2 =135 (270)/3 =90 Is z more or less than 90 degrees? How does z compare to x? W and Y look at least 90 degrees. X is more than 90 degrees.

11. Tactics 1 to 4: Dealing with Diagrams Tactic 3: If a Diagram is NOT Drawn to Scale, Redraw it, then trust your eyes.

12. WARNING: Note: Figure Not Drawn to Scale You cannot trust what you see in these pictures. For example, two lines may look perpendicular, but unless one of the labels is marked at 90 degrees, you cannot assume the lines form right angles.

13. (0,b) y = f(x) Example 3.1 The figure shows the graph of the linear function y = f(x). If the slope of the line is -2 and f(3) =4, what is the value of b? A)8 B) 9 C)10 D) 11 E) 12 Note: figure is not drawn to scale

14. There is a point on the graph at (3,4). What does f(3) = 4 mean? Sketch a coordinate plane, Use the point (3,4) and slope -2, Graph the line. B is the y-intercept (0,b) Where does the graph intersect the y-axis? A)8 B) 9 C)10 D) 11 E) 12

15. A 8 B X° C 4 Example 3.2 Hopefully, you can “eyeball” this angle and see it is 60 degrees. If you can’t, you can use that the leg is half the hypotenuse. This is only possible in the special right triangle 30-60-90. X is too big for 30, so it must be 60. • In triangle ABC, what is the value of x? A) 75 B) 60 C) 45 D) 30 E) 15 8 First decide a length for “4” Double it to make “8” Which answers are absurd? X° 4

16. To Redraw or not to redraw? Redrawing a picture can be a big waste of time. Only redraw a picture if you don’t see a straightforward solution to the problem. Redrawing takes time, accuracy, and ability. However, it may help you solve a problem you might otherwise skip.

17. Tactics 1 to 4: Dealing with Diagrams Tactic 4: Add a Line to the Diagram

18. A Line = Problem solving glasses Sometimes adding a line to a diagram can help you “see” a solution you wouldn’t have without it. Special lines can break up unusual shapes into familiar areas, help you see applications of Pythagorean Theorem, or see equivalent pieces of a diagram.

19. Q P O R Example 4.1 • the figure, Q is a point on the circle whose center is O and whose radius is r, and OPQR is a rectangle. What is the length of diagonal PR? A) r B) r2 C) r2 ÷π D) r√2 ÷π E) cannot be determined Any time you have a circle, try drawing in a radius. Diagonals of a rectangle are congruent. PR = OQ = r

20. B(1,7) C(11,5) A(1,1) D(5,1) Example 4.2 What is the area of quadrilateral ABCD? T 1 Add a line to break the shape into two triangles for which we can find the base and height. T 2 Triangle 1 Base = 6 Height = 10 Area = ½ (6)(10) Area = 30 Triangle 2 Base = 4 Height = 4 Area = ½ (4)(4) Area = 8 Total Area = 38

21. A O B P Example 4.3 In the figure, A and B are points on circle O and ray PA and ray PB are tangent to the circle. If m angle APB = 45, what is the degree measure of AOB (not shown)? Draw in the radii A radius and a tangent line always form right angles. A quadrilateral can be broken into 2 triangles. 2(180) = 360 total degrees 135 90 + 90 + 45 + AOB = 360 AOB = 135

22. IN CONCLUSION WHEN WORKING WITH DIAGRAMS— Don’t be afraid to draw a picture, add lines, redraw pictures, and trust what you see. Remember– the SAT has no complex geometry proofs… but, you may have to remember key geometry facts to reason through problems and find the right answers.