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# Tutor profile: Christine Q.

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Christine Q.
Exceptional Tutor who gets Exceptional Results! Academic, ACT, SAT, ISEE, SSAT
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## Questions

### Subject:SAT

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Question:

The coming decades will likely see more intense cluttering of jobs, innovation, and productivity in a smaller number of bigger cities and city-regions. Some regions could end up bloated beyond the capacity of their infrastructure, while others struggle, their promise sytmied by inadequate human or other resources. —Adapted from Richard Florida, the Great Reset Copy by Richard Florida. As used in line 55, “intense” most nearly means. A. emotional B. concentrated C. brilliant D. determined

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Christine Q.

### Subject:Geometry

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Question:

Draw a circle with center O and diameter AC. Line segment OA = 5 ft. Line segment OD reaches from the point O, at the center of the circle to point D at the edge of the circle and has a length of 5 ft. Line segment OB, also reaches from center O to the point B at the edge of the circle. Line BD is not a diameter of circle O, it is a chord of circle O. Chord BD is perpendicular to Diameter AC. Chord BD forms the third side of triangle OBD. Point E is the midpoint of line BD. Point E is collinear to points O and C (lies on line OC). Line segment EC has a length of 2. What is the length of BD?

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Christine Q.

This is a really fun question, because the key to geometry is always looking for circles and right triangles. Every shape we will deal with in Geometry includes circles and triangles—yes, including quadrilaterals, they are made up of triangles! We were given some very helpful measurements. The description is so specific that we must draw it out. The directions are so abundant, that we will have certainty that there are no mistakes. First let’s review some terms: Diameter: A line that spans the greatest width of a circle, it passes through the center of the circle. It is comprised of two radii. D = 2r. Radius: The distance from the center of a circle to the edge of the circle. Chord: Any line that connects one point on the edge of a circle to another point on the edge of a circle. A chord cannot cross through the circle center. A chord will never be longer than a diameter. Perpendicular Lines: Two lines that meet each other at a ninety degree angle. Now that we are clear with the vocabulary, let’s solve this question. 1. First: Let’s add any lengths that we can determine based on the givens in the problem. A radius is the distance from the center of a circle to the edge of the circle. We were not given the length of OB, but, based on the other givens OB must equal 5 because OB is a radius, just like OA and OD. This same radius situation applies to OC; it too must equal 5. 2. Now that we know that OC equals 5 and we know that EC equals 2, we know that OE = 3. 3. We have two legs of right triangles OED and OEB. We will use the Pythagorean theorem to find the length of ED. Since we are missing one of the legs of the right triangle, but we have the hypotenuse, we use the following equation: 3^2 + x^2 = 5^2 = 9 + x^2 = 25. 4. Solve for x using SADMEP, the opposite of PEMDAS. Subtract 9 from both sides of the equation to get: x^2 = 25 - 9, so x^2 = 16. 5. Let’s find the square root of both sides, of x^2 and of 16, to finish solving for x. x = 4 and x = -4. 6. Since this is a geometric shape, sides with a negative length are impossible. This means that x must be 4. Line ED = 4. 7. Line ED = Line EB because E is the given midpoint of line BD. 8. Therefore, the length of BD = 8.

### Subject:ACT

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Question:

Number 57 from ACT Math Test: Two numbers have a product of -48 and a sum of 0. What is the lesser of the 2 numbers? A. -4 rad 3 B. -3 rad 2 C. -2 rad 3 D. 0 E. 3

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Christine Q.

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