PSAT/SAT & ACT Math Attack – Geometry

Geometry Problem #1

Geometry Problem #1

The PSAT & SAT are notorious for asking complicated questions about seemingly simple concepts; this is one reason the PSAT & SAT Math sections can be so difficult. Following College Compass’ SAT Math Attack series will help you understand the concepts behind the questions.

First, identify the goal: what is the length of QS? Next, ask yourself, “how can I find the length of QS using the information in the problem?”  We can see that there are two right triangles in the image, so we can probably use the Pythagorean Theorem, which is used to find the side lengths of right triangles.  In order to find QS, we need to know the lengths of the other two sides — the problem gives us the length of QP, and it gives us information about the length of PR, but that’s not quite what we want.  Therefore, our new immediate goal is to find the length of PS.

Geometry problems are basically all about rules and how well you remember them.  For this problem, the main geometric principle that you need to know is: “a line that bisects an angle in a triangle also bisects the side opposite that angle.”

Because the lengths of QP and QR are equal, we know that this triangle is isosceles; thus, angles P and R are also equal.  If we view the triangle as two right triangles sitting side by side, we can see that they are identical — they both have right angles, they both have an angle equal to that of P and R, so their third angles must also be equal.  Therefore, line segment QS is proven to be a bisector of angle Q.  Since QS bisects angle Q, it must also bisect line segment PR.  This means that the length of segment PS is 12/2 = 6.  With this information, we can use the Pythagorean Theorem to find the length of QS.

a2 + b2 = c2
62 + b2 = 102
36 + b2 = 100
b2 = 64
b = 8

We could also have determined the length of segment QS without using the Pythagorean Theorem by recognizing the presence of a Pythagorean Triple: 6, 8, 10.

The answer is (C) because the two values are equal.

Geometry Problem #2

Geometry Problem #2

Now here’s an example of a question that yours truly got wrong the first time around.  There’s an important note in the directions of the math section that I didn’t read that you absolutely must know before doing any geometry problem:

Figures that accompany questions are intended to provide information useful in answering the questions.  However, unless a note states that a figure is drawn to scale, you should solve these problems NOT by estimating sizes by sight or by measurement, but by using your knowledge of mathematics.

Basically, if it does not say “Figure drawn to scale” (or something equivalent, like “RSTV is a square”), DO NOT ASSUME ANYTHING ABOUT THE FIGURE.

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The first time I did this problem, I glanced at it, thought it was an obtuse triangle, and answered (B).  However, because the problem does not specifically give information about the angles, that assumption was wrong.  This triangle could have been a (really, really, really) poorly drawn right triangle, in which case the answer would be (C).  If angle S is a right angle, then the problem is basically a statement of the Pythagorean Theorem, so RS2 + ST2 = RT2.  If it’s not a right triangle, then RT2 is NOT equal to RS2 + ST2 because the Pythagorean Theorem is ONLY true for right triangles; depending on the value of angle S, RT2 would be bigger or smaller.

The answer is (D) because there is not enough information given in the problem to determine the relationship.

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