SparkNotes_ SAT Subject Test_ Math Level 2_ Triangles.pdf

8/14/2019 SparkNotes_ SAT Subject Test_ Math Level 2_ Triangles.pdf

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8/12 SparkNotes: SAT Subject Test: Math Level 2: Triangles

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Plane Geometry

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6.1 LINES AND ANGLES 6.2 TRIANGLES 6.3 POLYGONS

6.4 CIRCLES 6.5 KEY FORMULAS

Triangles You will need a solid understanding of triangles in order to answer other questionsabout polygons, coordinate geometry, and trigonometry. Luckily for you, theessential rules governing triangles are few and easy to master.

Basic PropertiesThere are four main rules of triangles:

1. Sum of the Interior AnglesIf you were stranded on a desert island and had to take the Math IIC test, this is theone rule about tr iangles you should bring along: the sum of the measures of theinterior angles is 180. Now, if you know the measures of two of a triangles angles, you will be able to find the third. Helpful rule, dont you think?

2. Measure of an Exterior Angle An exterior angle of a triangle is the angle formed by ex tending one of the sides of thetriangle past a vertex (the point at which two sides meet). An exterior angle is alwayssupplementary to the interior angle with which it shares a vertex and equal inmeasure to the sum of the measures of the remote interior angles. Take a look at thefigure below, in which d , the exterior angle, is supplementary to interior angle c:

It doesnt matter which side of a triangle you extend to create an exterior angle; theexterior angle will always be supplementary to the interior angle with which it sharesa vertex and therefore (because of the 1 80 rule) equal to the sum of the remoteinterior angles.

3. Triangle Inequality The third important property of triangles is the triangle inequality rule, which states:the length of a side of a triangle is less than the sum of the lengths of the other twosides and greater than the difference of the lengths of the other two sides.

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Observe the figure below:

From the triangle inequality, we know that c b < a < c + b . The exact length of side adepends on the measure of the angle created by sides b and c. If this angle is large

(close to 1 80), then a will be large (close to b + c ). If this angle is small (close to 0),then a will be small (close to b c).

For an example, take a look at this triangle:

Using the triangle inequality, we can tell that 9 4 < x < 9 + 4, or 5 < x < 13. Theexact value of x depends on the measure of the angle opposite side x .

4. Proportionality of TrianglesThis brings us to the last basic property of triangles, which has to do with therelationships between the angles of a triangle and the lengths of the triangles sides. Inevery triangle, the longest side is opposite the largest angle and the shortest side isopposite the smallest angle.

In this figure, side a is clearly the longest side and is the largest angle. Converse ly,side c is the shortest side and is the smallest angle. It follows, therefore, that c < b< a and C < B < A. This proportionality of side lengths and angle measures holds true

for all triangles.

Special TrianglesThere are several special triangles that have particular properties. Knowing thesetriangles and what makes each of them special can save you time and effort.

But before getting into the different types of special triangles, we must take a momentto explain the markings we use to describe the properties of each particular triangle.For example, the figure below has two pairs of sides of equal length and threecongruent angle pairs: these indicate that the sides have equal length. The arcs drawninto A and B indicate that these angles are congruent. In some diagrams, theremight be more than one pair of equal sides or congruent angles. In this case, doublehash marks or double arcs can be drawn into a pair of sides or angles to indicate thatthey are equal to each other, but not necessarily equal to the other pair of sides orangles.

Now, on to the special triangles.

Scalene Triangles A scalene triangle has no equal sides and no equal angles.


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In fact, the special property of scalene triangles is that they dont really have any sp-ecial properties. Scalene triangles almost never appear on the Math IIC.

Isosceles Triangles A triangle that contains two sides of equal length is called an isosceles triangle. In anisosceles triangle, the two angles opposite the sides of equal length are congruent.These angles are usually referred to as base angles. In the isosceles triangle below,side a = b and A= B.

There is no such thing as a triangle with two equal sides and no congruent angles, or vice versa. From the proportionality rule, if a triangle has two equal sides, then thetwo angles opposite those sides are congruent, and if a triangle has two congruentangles, then the two sides opposite those angles are equal.

Equilateral Triangles A triangle whose sides are all of equal length is called an equilateral triangle. All threeangles in an equilateral triangle are congruent as well; the measure of each is 60 .

As is the case with isosceles triangles, if you know that a triangle has either threeequal sides or three congruent angles, then you know that the other must also be true.

Right Triangles A triangle that contains a right angle is called a right triangle. The side opposite theright angle is called the hypotenuse of the right triangle, and the other two sides arecalled legs. The angles opposite the legs of a right triangle are complementary.

In the figure above, C 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.

The Pythagorean TheoremThe Pythagorean theorem is vital to most of the problems on right triangles. It willalso come in handy later on as you study coordinate geometry and trigonometry. Thetheorem states that in a right triangle a 2 + b2 = c2 , where c is the length of thehypotenuse, a and b are the lengths of the two legs, and the square of the hypotenuseis equal to the sum of the squares of the two legs.

Pythagorean Triples


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Because right triangles obey the Pythagorean theorem, only a few have side lengthsthat are all integers. For example, a right triangle with legs of length 3 and 5 has ahypotenuse of length = 5.83.

The f ew sets of three integers that do obey the Pythagorean theorem and cantherefore be the lengths of the sides of a right triangle are called Pythagorean triples.Here are some common triples:

{3, 4, 5}

{5, 12, 13}

{7, 24, 25}

{8, 15, 17}

In addition to these Pythagorean triples, you should also watch out for theirmultiples. For example, {6, 8, 10} is a Pythagorean triple since its a multiple of {3, 4,5}.

Special Right TrianglesRight triangles are pretty special in their own right. But there are two extra -specialright triangles that appear frequently on the Math IIC. They are 30-60-90 trianglesand 45-45-90 triangles.

30-60-90 Triangles A 30-60-90 triangle is a triangle with angles of 30, 60, and 90. What makes itspecial is the specific pattern that the lengths of the sides of a 30-60-90 trianglefollow. Suppose the short leg, opposite the 30 angle, has length x . Then thehypotenuse has length 2 x , and the long leg, opposite the 60 degree angle, has length x . The sides of every 30-60-90 triangle will follow this 1 : 2 : ratio.

The constant ratio in the lengths of the sides of a 30-60-90 triangle means that if youknow the length of one side in the triangle, you immediately know the lengths of allthe sides. If, for example, you know that the side opposite the 30 angle is 2 meterslong, then by using the 1 : 2 : ratio, you know that the hypotenuse is 4 meters longand the leg opposite the 60 angle is 2 meters. On the Math IIC you will quite oftenencounter a question that will present you with an unnamed 30-60-90 triangle,allowing you to use your knowledge of this special triangle. You could solve thesequestions by using the Pythagorean theorem, but that method takes a lot longer thansimply knowing the proper 30-60-90 ratio. The key is to be aware that there are 30-60-90 triangles lurking out there and to strike when you see one.

45-45-90 Triangles A 45 -45-90 tr iangle is a triangle with two 45 angles and one right angle. This type of

triangle is also known as an isosceles right triangle, since its both isosceles and right.Like the 30-60-90 triangle, the lengths of the sides of a 45-45-90 triangle follow aspecific pattern that you should know. If the legs are of length x (they are alwaysequal), then the hypotenuse has length x . Take a look at this diagram:

As with 30-60-90 triangles, knowing the ratio for 4 5-45-90 triangles can save you agreat deal of time on the Math IIC.


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Similar TrianglesTwo triangles are called similar if the ratio of the lengths of their corresponding sidesis constant. In order for this to be true, the corresponding angles of each trianglemust be congruent. In essence, similar triangles have exactly the same shape, but notnecessarily the same size. Take a look at a few similar triangles:

As you may have assumed from the above figure, the symbol for is similar to is ~.So if triangle ABC is similar to tr iangle DEF , you could write ABC ~ DEF .

When you say that two triangles are similar, it is important to know which sides of each triangle correspond to each other. After all, the definition of similar triangles isthat the ratio of the lengths of their corresponding sides is constant. So, consideringthat ABC ~ DEF , you know that the ratio of the short sides equals the ratio of thelarger sides. AB / DE = BC / EF = CA / FD .

Just as similar triangles have corresponding sides, they also have congruent angles. If ABC ~ DEF , then A = D, B = E , and C = F .

Area of a TriangleThe formula for the area of a triangle is:

where b is the length of a base of the triangle and h is the height (also called thealtitude).

In the previous sentence we said a base instead of the base because you can

actually use any of the three sides of the triangle as the base; a triangle has noparticular side that is the base until you designate one. The height of the triangledepends on the base, which is why the area formula always works, no matter whichside you choose to be the base. The heights of a few triangles are pictured with theiraltitudes drawn in as dotted lines.

Study the triangle on the right. The measure of its height does not lie in the interior of the triangle. The height of a triangle is defined as a line segment perpendicular to theline containing the base , and not just the base. Sometimes the endpoint of the heightdoes not lie on the base; it can be outside of the triangle, as is the case in the right-most triangle in the figure above.

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SparkNotes_ SAT Subject Test_ Math Level 2_ Triangles.pdf