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Lessons On Punch Of the Week Every week, we come out with a set of questions that will help you gauge your level of preparation. Just remember - no cheating :) SAT:Punch Of the Week : 01-Nov-09 Ne...

Triangles

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:

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 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 , *d *must equal *a* + *b*.

3. Triangle Inequality Rule

This rule states that:

Take a look at the figure below:

The triangle inequality rule says that
*c* â€" *b* <
*a and a* < *c* + *b*. The exact
length of side *a*
depends on the measure of the angle created by sides *b* and *c*.

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 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.

Special Triangles

Isosceles Triangles

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
:

Equilateral Triangles

An equilateral triangle has three equal
sides and three congruent 60Âº
angles.

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 .

The Pythagorean
Theorem

If
triangles are

considered a SAT favorite then right triangles and the Pythagorean Theorem are

SAT darlings, so you jolly well know a lot about these two. Here's the Pythagorean

Theorem: In a right triangle,*a*^{2} +
*b*^{2} = hypotenuse^{2 }:

considered a SAT favorite then right triangles and the Pythagorean Theorem are

SAT darlings, so you jolly well know a lot about these two. Here's the Pythagorean

Theorem: In a right triangle,

PYTHAGOREAN TRIPLES

There
are few predefined sets of integers that always end up being what
we call "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 these, so you must
learn a few, to cut down the time you waste on calculations.

Extra-Special Right Triangles

Right triangles are pretty special in
their own way. 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.

30-60-90 Triangles

The guy who name the
30-60-90 triangle didn't have much of an imagination. As the name
clearly suggests, these triangles have angles of , , and . Before you say "So
what!" here's what's special: The sides of every 30-60-90 triangle will follow this
ratio of 1: : 2 . the diagram
explains this better:

This constant ratio means that if you
know the length of *just one* side in the triangle, you'll
immediately be able to calculate the lengths of all the
sides.

And there's another amazing thing about
30-60-90 triangles. Two of
these triangles joined at the side opposite the 60Âº angle will form an equilateral
triangle.

Here's why you need to pay attention to
this extra-special feature of 30-60-90 triangles. If you know the
side length of an equilateral triangle, you can figure out the
triangle's height: Divide the side length by two and multiply it by
.

45-45-90 Triangles

A
45-45-90 triangle is a
triangle with two angles of 45Â° and one right angle. It's
sometimes called an isosceles right triangle, since it's both
isosceles and right. If the legs are of length *x* (the legs will always be
equal), then the hypotenuse has length *x*:

Know
this 1: 1: ratio for 45-45-90 triangles. It will save you
time. Also, just as two 30-60-90 triangles form an equilateral
triangles, two 45-45-90
triangles form a square.

Similar Triangles

Two triangles are â€œsimilar" if the
ratio of the lengths of their corresponding sides is constant
(which you now know means that their corresponding angles must be
congruent). Take a look at a few similar triangles and you will
learn that the symbol for â€œis similar to" is ~ :

There
are two crucial facts about similar triangles.

- Corresponding sides of similar triangles are proportional.
- Corresponding angles of similar triangles are identical.

Congruent
Triangles

Congruent triangles are
identical.

Two triangles are congruent if they meet
any of the following criteria:

- All the corresponding sides of the two triangles are equal. This is known as the Side-Side-Side (SSS) method of determining congruency.
- The corresponding sides of each triangle are equal, and the mutual angles between those corresponding sides are also equal. This is known as the Side-Angle-Side (SAS) method of determining congruency .
- The two triangles share two equal corresponding angles and also share any pair of corresponding sides. This is known as the Angle-Side-Angle (ASA) method of determining congruency

Perimeter of a Triangle

The perimeter of a triangle is equal to
the sum of the lengths of the triangle's three sides.

Area of a Triangle

The formula for the area of a triangle
is

where *b* is the length of a base of
the triangle, and *h* is height (also called the
altitude). The height of the triangle will sometimes be shown as a
dotted line, as in the figures below:

But you probably won't get such an easy
question. It's more likely that you'll have to find the altitude,
using other tools and techniques. For example, try to find the area
of the triangle below:

To find
the area of this triangle, draw in the altitude from the base (of
length 9) to the opposite
vertex. Notice that now you have two triangles, and one of them
(the smaller one on the right) is a 30-60-90 triangle.

The hypotenuse of this 30-60-90 triangle is 4, so according to the ratio
1: : 2, the short side
must be 2 and the medium
side, which is also the altitude of the original triangle, is
2. *Now *you can
plug the base and altitude into the formula to find the area of the
original triangle: ^{1}/ _{2}*bh* =
^{1}/_{2}(9)(2) = 9.

Trig or Treat?

â€œOh my gosh! The new SAT includes trigonometry!" If you've heard people
talking this, don't listen to it. Here's what the actual
SAT people say about trig questions on
the new SAT: â€œThese questions can be
answered by using trigonometric methods, but may also be answered
using other methods." You will never *have* to use trig to
solve a problem, and I'll come right out and say it: You
*never* should use trig. That's right. At least not until
you are a genius at it!

The questions on which you could (but
shouldn't) use trig on the new SAT will
cover 30-60-90 and
45-45-90 triangles.

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There are other formulas as well to find the area of triangles. Bec they never ask area based of ht base formula.

what about the rest of SOT i mean bout-sine/cosine rules,areas of triangles inscribed in certain geometrical figures as circles ,squares?? area of circumcircle/incircle their centres etc..aren't they asked much in SAT??

very nice lesson but dont u think in the asa method of determining congruency, the congruent sides should be in between the 2 congruent angles?im not quite sure about this , anyways nice work. :)

Aarushi you are right… actually they are two separate rules but for convenience i merged both! The property is as follows: The triangle can have 2 corresponding angles equal and the side in between them equal or any other side equal. For further clarification look up these links: http://www.mathopenref.com/congruentaas.html.

Technically, if you know 2 angles you can figure out the third in one second, so it wouldn't really matter where the side lies, in between or not. You do have to make sure the side makes sense though, but this can be done easily by noting its direction relative to the angles given.

Incredible lesson Astha! Keep up the fantastic work!

WOW! Great lesson Astha! Very excited to see more. :)

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