# Multiple Figures

## Geometry: Multiple Figures

Read the following SAT test question and then select the correct answer.

Use the same steps for every math problem.  First, read the question carefully and identify the bottom line.  Next, assess your options and choose the most efficient method to attack the problem.  Finally, loop back to verify that your answer addresses the bottom line.

In the figure above, if PQRS is a quadrilateral and TUV is a triangle, what is the sum of the degree measures of the marked angles?

Bottom Line:  Sum of degrees of the marked angles = ? (Write Sd = ?)

Assess your Options:  You could try to find the individual angles, but you don’t have enough information to do this.  Instead, use the rules you have memorized about each shape.

Attack the Problem:  You know that TUV is a triangle.  All the angles of a triangle add up to 180 degrees.  You know that PQRS is a quadrilateral.  All the angles of a quadrilateral add up to 360 degrees.  In the image, you can see that all of these angles in each of these two shapes are marked, and you know that you are looking for a sum, so add them together.  180 + 360 = 540.

Loop back: Your answer is in degrees and you have found the total of all the marked angles.  Look down at your answer choices.

(A) 420
(B) 490
(C) 540
(D) 560
(E) 580

On sat.collegeboard.org, 77% of the responses were correct.

For more help with SAT writing, visit www.myknowsys.com!

Read the following SAT test question and then select the correct answer.

Math questions should always be read carefully.  You will also avoid making errors by identifying the bottom line and assessing your options for solving the question.  Choose the most efficient method to attack the problem.  When you have finished, loop back to be sure that even if there were multiple steps, you reached the bottom line.

The length of a rectangle is increased by 20%, and the width of the rectangle is increased by 30%. By what percentage will the area of the rectangle be increased?

Bottom Line: % change = ?

Assess your Options:  You could work this problem without picking any numbers; however, picking easy numbers will allow you to think about the problem in a more concrete way and avoid errors.

Attack the Problem:  One of the easiest numbers to work with is one.  Think of your original rectangle as having a length of one and a width of one.  The formula for area of a rectangle is length times width.  If L × W = A, for your first rectangle you have 1 × 1 = 1.   The area of the original rectangle is one.

Then think about the changes that occur to that rectangle.  The length increases by 20%.  In order to find 20% of 1, all you have to do is move the decimal over twice to .2.  The new length is 1.2.  Use the same method to find the new width, and an increase of 30% becomes 1.3.  The area of the rectangle after the change is 1.2 × 1.3 = 1.56

The formula for percent change would require you to find the difference between these two areas and divide that by the original number.  You use the same formula whether you are looking for an increase or a decrease.  Notice that your original number is one, so dividing by one will not change your answer.  All you need to do is find the difference between the areas: 1.56 – 1 = .56.  What is .56 as a percent?  Your answer is 56%.

Loop Back:  You found the percent change, which was your bottom line.  Look down at your answer choices.

(A) 25%
(B) 36%
(C) 50%
(D) 56%
(E) 60%

On sat.collegeboard.org, 34% of the responses were correct.

For more help with SAT math, visit www.myknowsys.com!

Remember the Knowsys Method: Without looking at the answer choices, read the question carefully, note the bottom line, assess your options, attack the problem, and loop back to check that you found what the question wanted.

What is the maximum number of nonoverlapping squares with sides of length 3 that will fit inside a square with sides of length 6?

At the top of your scratch work, summarize "the maximum number of nonoverlapping squares":

max squares = ?

Next, assess your options. There are two good ways to solve this quadrilaterals problem: visual and mathematical. Those who learn and think more visually can sketch or imagine a square with side length of 6, then reason that each side would be cut in half to make squares with sides of length 3. Two small squares touch each side of the large square, so four small squares total fit into the larger square.

Alternately, you can calculate the area of the large square and divide it among the smaller squares. The large square has sides of length 6, so its area is  $6\ast 6= 36$. You will also need the area of the small squares. Their area is $3\ast 3= 9$. Finally, divide the area of the large square by the area of the small square to determine how many will fit in the larger square. $36\div 9=4$

Both methods gave 4. Now look at the answer choices:

A) Two
B) Three
C) Four
D) Six
E) Nine