Geometry, SAT – angle relationships

Q: In the diagram below, what is the value of x?
 
 
198930_185618174816861_836602_n
 
 
Answer: 40
Explanation: The interior angle adjacent to 70 is 110. The other interior angle is 30 (from 180-150). Together, these two interior angles are 140. Subtract from 180.  The third angle of the triangle is 40.  x is a vertical angle to the 40, so it is also 40.
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Geometry – polygons

Q: What is the sum of the interior angles of a convex hexagon?
 
 Answer: 720
Use the formula (n-2)x180, with n=6 (because a hexagon has 6 sides).
(6 – 2) x 180
4 x 180
720
My TI-84 Plus program POLYGON will find this total for you.
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Geometry – Circles

Q: The radius of Circle M is twice as long as the radius of Circle N. What is the ratio of the area of Circle M to the area of Circle N?
 
Answer:  4:1 or 4/1. 
Explanation:  The ratio of the radii (2:1) is a length ratio. Square the length ratio to get the area ratio (4:1).  This works for all pairs of similar figures.
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Algebra 1 – distance

Q: To the nearest tenth, what is the distance between the points (-6,2) and (6,5)?
 
Answer:  12.4 
(Use the distance formula, and round to the nearest tenth.)
My TI-84 Plus program DISTANCE will find this value.
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Geometry – Volume

 

Q: What is the maximum number of boxes with dimensions 4”x5”x8” that can be packed into one large box with dimensions 20”x20”x24”?
Answer:  60

Explanation: First, decide that a whole number of small boxes will entirely fill the large box.  To do this, look at the dimensions and match them.  4″ fits into 20″ five times.  5″ fits into 20″ four times.  8″ fits into 24″ three times.  That means, you’ll get five rows of small boxes, four boxes per row, and three layers deep.  So, the quickest way to get the answer is to multiply 5x4x3 = 60.

BUT if you don’t notice that, just do (large volume)/(small volume).

Large volume = 20x20x24 = 9600

Small volume = 4x5x8 = 160

(large volume)/(small volume) = 9600/160 = 60

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Geometry – Volume

Q: Two rectangular prisms have the same volume. Prism 1 has dimensions 4 ft, 5 ft, and 7 ft. Prism 2 has dimensions 14 ft, 4 ft, and x ft. What is the value of x?
Answer:  2.5

Explanation:  Volume of a rectangular prism = (L)(W)(H)

Volume 1 = (4)(5)(7) = 140

Volume 2 = (14)(4)(x) = 56x

Since the volumes are equal, 56x = 140.  x = 2.5

 

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Geometry – circles

Q: The area of a 45 degree sector of circle M is 100 square units. What is the area of circle M?

Answer: 800 square units

Explanation:  The 45° sector is exactly 1/8 of the circle. (45/360 = 1/8)

100 = 1/8 of the area of the whole circle.

The area of the whole circle is 800.

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Geometry – prism

 How many vertices does a rectangular prism have?


Answer:   8
rectangularprismvertices

If you have trouble drawing it or visualizing it, consider this instead:  A prism has two identical bases.  All of the vertices are along the edges of the two bases.  Since a rectangular prism has rectangles for bases, each base will have 4 vertices.  Two rectangular bases will have 8 vertices total.

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Algebra 2 – solving quadratic equations

Q: The length of a rectangle is 4 cm longer than the width. If the rectangle’s area is 45 cm², what is the width?

Answer: 5 cm

Explanation:  Call the width x.  Then, the length is x+4.

Area of a rectangle = (length)(width).

45 = x(x+4)

45 = x² + 4x

0 = x² + 4x – 45

Now, factor:

0 = (x + 9)(x – 5)

x = -9 or x = 5

Since a length can’t be negative, discard x = -9.  The only solution is x = 5.  Since ‘x’ is what we called the width, it is the answer to the question.  Just put the appropriate unit on it: 5 cm.

 

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