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SAT 数学对三角形的全面解析
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Triangles pop up all over the Math section. There are questions specifically about triangles,
questions that ask about triangles inscribed in polygons and circles, and questions about
triangles in coordinate geometry.
Three Sides, Four Fundamental Properties
Every triangle, no matter how special, follows four main rules.
1. Sum of the Interior Angles
If you were trapped on a deserted island with tons of SAT questions about triangles, this
is the one rule you’d need to know:
The sum of the interior angles of a triangle is 180°.
If you know the measures of two of a triangle’s angles, you’ll always be able to find
the third by subtracting the sum of the first two from 180.
2. Measure of an Exterior Angle
The exterior angle of a triangle is always supplementary to the interior angle with which
it shares a vertex and equal to the sum of the measures of the remote interior angles. An exterior
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angle of a triangle is the angle formed by extending one of the sides of the triangle past a
vertex. In the image below, d is the exterior angle.
Since d and c together form a straight angle, they are supplementary:
. According to the first rule of triangles, the three angles of a triangle always add up
to
, so
. Since
and
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, d must equal a + b.
3. Triangle Inequality Rule
If triangles got together to write a declaration of independence, they’d have a tough time,
since one of their defining rules would be this:
The length of any side of a triangle will always be less than the sum of the lengths of
the other two sides and greater than the difference of the lengths of the other two sides.
There you have it: Triangles are unequal by definition.
Take a look at the figure below:
The triangle inequality rule says that c – b < a < c + b. The exact length of side a depends
on the measure of the angle created by sides b and c. Witness this triangle:
Using the triangle inequality rule, you can tell that 9 – 4 < x < 9 + 4, or 5 < x < 13.
The exact value of x depends on the measure of the angle opposite side x. If this angle is large
(close to
) then x will be large (close to 13). If this angle is small (close to
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), then x will be small (close to 5).
The triangle inequality rule means that if you know the length of two sides of any triangle,
you will always know the range of possible side lengths for the third side. On some SAT triangle
questions, that’s all you’ll need.
4. Proportionality of Triangles
Here’s the final fundamental triangle property. This one explains the relationships between
the angles of a triangle and the lengths of the triangle’s sides.
In every triangle, the longest side is opposite the largest angle and the shortest side
is opposite the smallest angle.
In this figure, side a is clearly the longest side and
is the largest angle. Meanwhile, side c is the shortest side and
is the smallest angle. So c < b < a and C < B < A. This proportionality of side lengths
and angle measures holds true for all triangles.
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See if you can use this rule to solve the question below:
What is one possible value of x if angle C < A < B?
(A)1
(B)6
(C)7
(D)10
(E)15
According to the proportionality of triangles rule, the longest side of a triangle is opposite
the largest angle. Likewise, the shortest side of a triangle is opposite the smallest angle.
The largest angle in triangle ABC is
, which is opposite the side of length 8. The smallest angle in triangle ABC is
, which is opposite the side of length 6. This means that the third side, of length x, measures
between 6 and 8 units in length. The only choice that fits the criteria is 7. C is the correct
answer.
Special Triangles
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Special triangles are “special” not because they get to follow fewer rules than other
triangles but because they get to follow more. Each type of special triangle has its own special
name: isosceles, equilateral, and right. Knowing the properties of each will help you tremendously,
humongously, a lot, on the SAT.
But first we have to take a second to explain the markings we use to describe the properties
of special triangles. The little arcs and ticks drawn in the figure below show that this triangle
has two sides of equal length and three equal angle pairs. The sides that each have one tick
through them are equal, as are the sides that each have two ticks through them. The angles with
one little arc are equal to each other, the angles with two little arcs are equal to each other,
and the angles with three little arcs are all equal to each other.
Isosceles Triangles
In ancient Greece, Isosceles was the god of triangles. His legs were of perfectly equal
length and formed two opposing congruent angles when he stood up straight. Isosceles triangles
share many of the same properties, naturally. An isosceles triangle has two sides of equal length,
and those two sides are opposite congruent angles. These equal angles are usually called as
base angles. In the isosceles triangle below, side a = b and
:
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If you know the value of one of the base angles in an isosceles triangle, you can figure
out all the angles. Let’s say you’ve got an isosceles triangle with a base angle of 35º. Since
you know isosceles triangles have two congruent base angles by definition, you know that the
other base angle is also 35º. All three angles in a triangle must always add up to 180º, right?
Correct. That means you can also figure out the value of the third angle: 180º – 35º – 35º
= 110º.
Equilateral Triangles
An equilateral triangle has three equal sides and three congruent 60º angles.
Based on the proportionality rule, if a triangle has three equal sides, that triangle must
also have three equal angles. Similarly, if you know that a triangle has three equal angles,
then you know it also has three equal sides.
Right Triangles
A triangle that contains a right angle is called a right triangle. The side opposite the
right angle is called the hypotenuse. The other two sides are called legs. The angles opposite
the legs of a right triangle are complementary (they add up to 90º).
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In the figure above,
is the right angle (as indicated by the box drawn in the angle), side c is the hypotenuse,
and sides a and b are the legs.
If triangles are an SAT favorite, then right triangles are SAT darlings. In other words,
know these rules. And know the Pythagorean theorem.
The Pythagorean Theorem
The Greeks spent a lot of time reading, eating grapes, and riding around on donkeys. They
also enjoyed the occasional mathematical epiphany. One day, Pythagoras discovered that the sum
of the squares of the two legs of a right triangle is equal to the square of the hypotenuse.
“Eureka!” he said, and the SAT had a new topic to test.
Here’s the Pythagorean theorem: In a right triangle, a2 + b2 = c2:
where c is the length of the hypotenuse and a and b are the lengths of the two legs.
The Pythagorean theorem means that if you know the measures of two sides of a right triangle,
you can always find the third. “Eureka!” you say.
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Pythagorean Triples
Because right triangles obey the Pythagorean theorem, only a specific few have side lengths
that are all integers. For example, a right triangle with legs of length 3 and 5 has a hypotenuse
of length
= 5.83.
The few sets of three integers that do obey the Pythagorean theorem and can therefore be
the lengths of the sides of a right triangle are called Pythagorean triples. Here are some common
ones:
{3, 4, 5}
{5, 12, 13}
{7, 24, 25}
{8, 15, 17}
In addition to these Pythagorean triples, you should also watch out for their multiples.
For example, {6, 8, 10} is a Pythagorean triple, since it is a multiple of {3, 4, 5}.
The SAT is full of right triangles whose side lengths are Pythagorean triples. Study the
ones above and their multiples. Identifying Pythagorean triples will help you cut the amount
of time you spend doing calculations. In fact, you may not have to do any calculations if you
get these down cold.
Extra-Special Right Triangles
Right triangles are pretty special in their own right. But there are two extra-special right
triangles. They are 30-60-90 triangles and 45-45-90 triangles, and they appear all the time
on the SAT.
In fact, knowing the rules of these two special triangles will open up all sorts of time-saving
possibilities for you on the test. Very, very often, instead of having to work out the Pythagorean
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theorem, you’ll be able to apply the standard side ratios of either of these two types of triangles,
cutting out all the time you need to spend calculating.
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