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Geometry
More Geometry
A lot of geometry questions on the GRE pertain to triangles, so I want to cover more
material about them than I did in our last session.
Acute angle: An angle that is less than 90 degrees.
Obtuse angle: An angle that is more than 90 degrees.
Acute triangles: all three angles are acute (less than 90 degrees).
Obtuse triangles: One of the angles is obtuse and the other two are acute.
Right triangles: It has one right angle (90 degree angle) and two acute angles.
Classification of Triangles:
Length of
Name:
the sides
Measure of
the angles
Scalene
all 3 are different
all 3 are different
Isosceles
2 are the same
2 are the same
Equilateral
all 3 the same
all 3 the same
====================================================
One of the things that we mentioned last time was Pythagorean’s Theorem (which applies
ONLY to right triangles):
B
a
c
a2 + b2 = c2
C
b
A
1 Here are the relationships between a, b, and c for right, obtuse, and right triangles:
a2 + b2 = c2 if and only if angle C is a right angle (Triangle ABC is a right triangle).
a2 + b2 < c2 if and only if angle C is a obtuse (Triangle ABC is an obtuse triangle).
a2 + b2 > c2 if and only if angle C is an acute angle (Triangle ABC is an acute triangle).
Here are the right triangles that appear the most often on the GRE:
The 30o-60o-90o triangle:
The sides are x, x 3 , and 2x:
If you know the length of the shorter leg (x),
Multiply it by 3 to get the longer leg, and
Multiply it by 2 to get the hypotenuse.
If you know the length of the longer leg (x 3 ),
Divide it by 3 to get the shorter leg, and
Multiply the shorter leg by 2 to get the hypotenuse.
If you know the length of the hypotenuse (2x),
Divide it by 2 to get the shorter leg, and
Multiply the shorter leg by 3 to get the longer leg.
2 The 45o-45o-90 o triangle:
The sides are x, x, and x 2 :
By multiplying the length of a leg by 2 , you get the hypotenuse.
By dividing the hypotenuse by 2 , you get the length of each leg.
=================================================
In the diagram, if BC = 6 ,
what is the value of CD?
A. 2 2
B. 4 2
C. 2 3
D. 2 6
E. 4
Solution: Choice (E)
Triangle ABC is a 30o-60o -90o triangle, and you are given the longer leg, which you can
use to find the other two sides:
6
6
Divide the longer leg BC by 3 to get the shorter leg AB =
=
= 2
3
3
Multiply AB by 2 to get the hypotenuse AC = 2 2
Triangle DAC is a 45o-45o-90o triangle, and you now know the length of its leg AC.
To get the hypotenuse DC:
DC = 2 2 2 = (2)(2) = 4
Multiply AC by 2 :
( )( )
3 The measures of the three angles in a triangle are in the ratio of 1:1:2. Which of the
following statements must be true?
Indicate all such statements.
A. The triangle is isosceles
B. The triangle is a right triangle
C. The triangle is equilateral
Solution: Choices (A) and (B)
You could create a “Ratio Box” to help you organize your thoughts:
side 1
side 2
side 3
Total
ratio
1
1
2
4
multiply by
?
real
180o
We know that for a triangle, the sum of the angles is 180o.
4 would need to be “multiplied by” 45o in order to equal 180o
side 1
side 2
side 3
Total
ratio
1
1
2
4
multiply by
45o
real
180o
Now go ahead and multiply the others by 45o to see what each of the angles would have
to be:
side 1
side 2
side 3
Total
ratio
1
1
2
4
o
o
o
multiply by
45
45
45
45o
real
45o
45o
90o
180o
So this would be a 45o-45o-90o triangle. Since two of the angles have the same measure,
the triangle is isosceles, and since one of the angles is 90o, it is a right triangle.
=====================================================
In the last session we mentioned that the
area of a triangle is calculated using this formula:
A = 1 bh
2
4 DEFG is a rectangle. What is the area of triangle DFH?
A. 3
B. 4.5
C. 6
D. 7.5
E. 10
Solution: Choice (B)
Triangle DGH is a right triangle, so you can use Pythagorean’s Theorem to find the
length of the leg GH: 32 + (GH)2 = 52 → GH = 25 − 9 = 16 = 4
Since GF = DE = 7, HF = 7 – 4 = 3.
Triangle DFH has a base of HF = 3, and a height of DG = 3
1
1
So the area is A = bh = (3)(3) = 4.5
2
2
What is the perimeter of triangle ABC?
A. 48
B. 48 + 12 2
C. 48 + 12 3
D. 60
E. 60 + 6 3
Solution: Choice (C)
Triangle ABD is a right triangle, so you can use Pythagorean’s Theorem to solve for the
leg AD: AD = 152 − 92 = 225 − 81 = 144 = 12
Triangle ADC is a 30o-60o -90o triangle, whose shorter leg x is 12. The hypotenuse 2x is
AC = 24. And the other leg x 3 is CD = 12 3 .
Perimeter = 24 +15 + 9 + 12 3 = 48 + 12 3
5 Remember covering this last time?:
Know these relationships between the angles and sides of a triangle:
• The longest side is opposite the largest interior angle. The shortest side is opposite
the smallest interior angle. Equal sides are opposite equal angles.
• The length of any one side of a triangle must be less than the sum of the other two
sides and greater than the difference between the other two sides. So take any two
sides of a triangle, add them together, then subtract one from the other, and the
third side must lie between those two numbers.
The lengths of two sides of a triangle are 7 and 11.
Quantity A
Quantity B
The length of
the third side
A.
B.
C.
D.
4
Quantity A is greater
Quantity B is greater
Quantities A and B are equal.
It is impossible to determine which
quantity is greater.
Solution: Choice (A)
The length of the third side must be: 11-7 < third side < 7+11
4 < third side < 18
The third side is greater than 4.
Polygons
A polygon is a closed geometric figure made up of line segments.
The simplest polygon has three sides and is a triangle.
A polygon with four sides is a quadrilateral.
The only other ones you need to be familiar with are:
A polygon with five sides is a pentagon.
A polygon with six sides is a hexagon.
A polygon with eight sides is an octagon.
A polygon with ten sides is a decagon.
Every quadrilateral has two diagonals.
When you draw in either one, you divide the quadrilateral into two triangles.
Since the sum of the three angles in any triangle is 180o, the sum of the angles in any
quadrilateral has to be (2)(180o ) = 360o
6 This same process can be applied to any polygon, dividing it into triangles by drawing in
all of the diagonals that emanate from one vertex:
A Pentagon is divided into three triangles, so the sum of the angles in any pentagon has to
be (3)(180o ) = 540o
A Hexagon is divided into four triangles, so the sum of the angles in any hexagon has to
be (4)(180o ) = 720o
Is there a pattern here? Yes! Just count the number of sides, subtract 2, and then
multiply the result by 180o.
========================================================
What is the measure, in degrees, of each interior angle in a regular decagon?
Solution: 144o
A decagon has 10 sides, and two less than this is 8. Multiplying 8 by 180o tells us the
sum of all the angles is 1440o. All 10 angles equal each other, so divide 10 into 1440o to
find the size of each interior angle: 1440o ÷ 10 = 144o
A Parallelogram is a quadrilateral in which both pairs of opposite sides are parallel.
Opposite sides are equal.
Opposite angles are equal.
Consecutive angles add up to 180o
The two diagonals bisect each other.
The diagonal divides the parallelogram into two triangles that have the exact same size
and shape.
A Rectangle is a parallelogram, so the same properties above apply to it, plus these:
The measure of each angle in a rectangle is 90o.
The diagonals of a rectangle have the same length.
7 A Square is a rectangle, so the same properties for rectangles apply to it, plus these:
All four sides have the same length.
Each diagonal divides the square into two 45o-45o-90o right triangles.
The diagonals are perpendicular to each other.
A Trapezoid is a quadrilateral in which one pair of sides IS parallel and the other pair of
sides is NOT parallel. The parallel sides are called the bases.
Here are the Area Formulas you need to know:
Area for a Parallelogram: Area = (base)(height)
Area for a Rectangle: Area = (length)(width)
Area for a Square: Area = (side)2 also: Area =
Area for a Trapezoid: Area =
1 2
d
2
1
( base1 + base2 ) ( height )
2
Two rectangles with the same perimeter can have different areas.
Two rectangles with the same area can have different perimeters.
For a given perimeter, the rectangle with the largest area is a Square.
For a given area, the rectangle with the smallest perimeter is a Square.
8 Quantity A
The area of a
rectangle whose
perimeter is 12
A.
B.
C.
D.
Quantity B
10
Quantity A is greater
Quantity B is greater
Quantities A and B are equal.
It is impossible to determine which
quantity is greater.
Solution: Choice (B)
There are a lot of different areas whose perimeters are 12.
The largest area would be created by a 3x3 square, which is 9.
=======================================================
WXYZ is a parallelogram
Quantity A
Quantity B
Diagonal WY
Diagonal XZ
A.
B.
C.
D.
Quantity A is greater
Quantity B is greater
Quantities A and B are equal.
It is impossible to determine which
quantity is greater.
Solution: Choice (B)
Angle Z is an acute angle, so (WY)2 < a2 + b2
Angle Y is an obtuse angle, so (XZ) 2 > a2 + b2
So XZ > WY
9 More on Circles
circumference C
=
diameter
d
C = πd
C = 2π r
π is approximately 3.14 (a little more than 3).
The degree measure of a complete circle is 360o.
Area = π r 2
x
If an arc measures xo, the length of the arc is
( 2π r )
360
and the area of the sector formed by the arc and the two radii is
π=
( )
x
π r2
360
===========================================================
Each of the triangles is equilateral.
Quantity A
The area of the
shaded region
A.
B.
C.
D.
Quantity B
6π
Quantity A is greater
Quantity B is greater
Quantities A and B are equal.
It is impossible to determine which
quantity is greater.
10 Solution: Choice (C)
Because the triangles are equilateral, the two non-shaded central angles are 60o each
giving us a sum of 120o (imagine “combining” the two non-shaded regions together).
2
120 1
The white area is
= of the circle, so the shaded area is of the circle.
3
360 3
2
2
The area of the entire circle is π r = π 3 = 9π
2
The shaded area is two-thirds of this: (9π ) = 6π
3
Solid Geometry
Volume
The volume of a three-dimensional figure is found by multiplying the area of the twodimensional figure by the height.
Rectangular solid: Volume = (Area of a rectangle) × (depth) = length × width × height
Circular cylinder: Volume = (Area of a circle) × (height) = π r 2h
A diagonal of a box is the longest line segment that can be drawn between two points on
the box.
To find the length of the diagonal of a box use this formula:
a2 + b2 + c2 = d 2 where a, b, and c are the dimensions of the rectangular box, and d is
the length of the diagonal.
Just so you know, this formula is just an extension of
the Pythagorean Theorem.
EG is the diagonal of the rectangular base EFGH.
Since the sides of the base are 3 and 4, you can use
Pythagorean’s Theorem to find that EG is 5.
Now, triangle CGE is a right triangle whose legs
are 12 and 5, so you can use Pythagorean’s Theorem
on it to find that EG (the diagonal of the box) is 13.
So you could have solved it this way, but it’s a little
faster to use a2 + b2 + c2 = d 2
11 Surface Area
For a rectangular box the surface area is the sum of the areas of all of its sides.
For a cylinder the surface area is the sum of :
the area of its side: (circumference of the base)(height) = 2 π (radius)(height)
the area on the top: π (radius)2
the area on the bottom: π (radius)2
=====================================================
A 5-foot-long cylindrical pipe has an inner diameter of 6 feet and an outer diameter of 8
feet. If the total surface area (inside and out, including the ends) is k π , what is the value
of k?
A.
B.
C.
D.
E.
7
40
48
70
84
Solution: Choice (E)
Draw a sketch:
Surface area of a side of a cylinder is A = 2 π rh, so the area of the
exterior is 2 π (4)(5) = 40 π , and the area of the
interior is 2 π (3)(5) = 30 π .
The area of EACH shaded end is the area of the
outer circle minus the area of the inner circle:
π (4)2 - π (3)2
16 π - 9 π
7π
So the total surface area is:
40 π + 30 π + 7 π + 7 π = 84 π
Total Surface Area = k π = 84 π
k = 84
12