7.3 SAT Math - Triangles

Triangles are referred to by their three points, often with a triangle symbol in front, such as ΔJKL.  The three angles of a triangle always add up to 180°.

The perimeter of a triangle (like any polygon) is found by simply adding up the lengths of the three sides.
-SAT Math Tip: Triangles often have odd numbers for side lengths, things like 6√3 or even π/4.  Don't worry about it if they're weird, just add them up.  The answer choices will reflect the weirdness, so just look for the answer that matches what you got.  If your answer choice is in square root form and the answers are decimals, just type it into your calculator to convert and check yourself.

The area of a triangle is found through the formula , where b stands for the base and h for the height.  The height is also sometimes referred to as the altitude. 
-SAT Math Hint: Triangles have 3 bases and 3 heights!  To find the height from any base, drop a perpendicular line to the base from the vertex that is NOT part of the base.  Take a look:

Sometimes, in order to drop a perpendicular line, it will be necessary to extend the base out to the point where the perpendicular line would hit it.  This is always necessary for an obtuse angle:

Keep in mind that you are only extending the base to see where the perpendicular line drops!  The original base is STILL the number you use whenever you have to do work, not the extended base.

Right Triangle

A right triangle is a triangle in which one of its angles equals 90°.  Right triangles are special for a number of reasons, but the first regards bases and heights – in a right triangle, your base and your height are the two sides that meet in a 90° angle; it is not necessary to draw a new height.

In a right triangle, the two sides of the triangle that meet in a 90° angle are called the legs.  The side opposite the 90° angle is the hypotenuse.  If you're not sure which one is the hypotenuse, the corner of the 90° box always points to the hypotenuse:

If you know two sides of a right triangle, you can find the third through the Pythagorean Theorem:

a² + b² = c²

where a and b are the legs, and c is the hypotenuse.
-SAT Math Hint: A little-talked about fact about right triangles is that the hypotenuse is always the longest of the three sides.  This is a quick and easy way to narrow your SAT answer choices when you are given the hypotenuse – the legs might be very close in value to the hypotenuse, but they can neither equal it nor be greater than it.

Triangles share a relationship between their sides and angles.  The wider the angle, the longer the side opposite that angle will be.  The more acute the angle, the shorter the side opposite that angle will be.

This rule can be tested in problems like the one below.

Note: Figure not drawn to scale.

5. In triangle PQR above, if p = 58, q = 59, and r = 61, which of the following must be true?
(A) PR < QR < RP (B) PQ < PR < QR (C) QR < PQ < PR (D) QR < PR < PQ (E) PQ > QR > PR

Always know what answer you're looking for before you go to the answer choices.  In this case, you should write out the relationship between the sides first, then see which answer choice matches.  If you try each answer choice instead to see if it works, it will take you too long and get too confusing.

The answer is D).

Isosceles Triangle

An isosceles triangle has two equal sides.  As a result, it also has two equal angles.  The two equal angles will be the two angles that share one of the equal sides and one of the not equal sides:

Equilateral Triangle

An equilateral triangle has all three sides equal.  As a result, all three angles are also equal, which means every angle in an equilateral triangle equals 60°.
-SAT Math Hint: You don't have to know the following formula for this test, but if you have an equilateral triangle, there's a special formula for its area: , where s is a side.  If you don't have this formula memorized, you'll still be able to figure out the area by finding the height, just like you would any other triangle.

Scalene Triangle

A scalene triangle is simply a triangle in which none of the sides, and therefore none of the angles, equal each other.

Here's an odd fact about triangles that they sometimes test: Any two sides of a triangle must add up to a number that's greater than the third side.  This means that a triangle CANNOT have sides of 3, 3, and 7 for example, because it would look like one of these:

The most common way this rule will be tested is in a question like the following:
11. A triangle has sides of 8, 10, and x.  What are the possible values of x?

To find the possible sides, simply add the two known sides together to get the highest value possible, and subtract the two sides to get the lowest value possible. 
-SAT Math Tip: Don't forget that those two values represent the limits of the sides – which means the sides CANNOT actual equal either of those two numbers!  The sum of any two sides must be greater than the third, not equal to. 

If you add the two numbers, you get 18; if you subtract, you get 2.  So the possible values of the third side are 2 < x < 18.  Again, remember, x cannot equal 2 or 18, so if they ask how many possible integer values there are for x, the answer to that question would be 15.

Special Right Triangles

A 30-60-90 triangle has one 30° side, one 60° side, and one 90° side:

The shortest side is always the side opposite the 30° angle.  To find the side opposite the 60° angle, multiply the shortest side by √3.  To find the hypotenuse, multiply the shortest side by 2.
-SAT Math Hint: Always go through the shortest side.  It doesn't take that long to find it if you were given a different side, and it makes remembering the formulas and doing the work much much easier.

An isosceles right triangle is also known as a 45-45-90. 

Multiply either of the two sides by √2 to find the hypotenuse, or divide the hypotenuse by √2 to find a side.
-SAT Math Hint: Do NOT use special angles (30, 45, 60 or 90) when you are throwing your own numbers into a question, such as when you are using ITOS.  They will cause repeat answers and you will need to try new angles anyway.  If you need two angles to add up to 90°, try 40 and 50 or 35 and 55 instead.

Some common sides of right triangles are:
3, 4, 5 5, 12, 13 8, 15, 17 7, 24, 25

The multiples of these sides are also common, so 6, 8, 10 or 16, 30, 34 can also be common.  You do not need to have these memorized, but if you do, it can save you some work when you have to solve for a side. 
-SAT Math Hint: These are also good numbers to try when you are plugging your own numbers into a question, as you know for a fact that a triangle of lengths 3, 4, 5 does exist.  Throwing in random numbers for triangle sides can be dangerous, as you might accidentally pick sides of impossible triangles, which may ruin all your results.

Similar/Congruent Triangles

If all three angles of two triangles are equal, those two triangles are considered similar triangles.  This means that their sides will be in proportion to each other.  Alternatively, if two triangles have all the sides in proportion to each other, all of their corresponding angles are equal.

If ∠A = ∠D, ∠B = ∠E, and ∠C = ∠F, then all three sides are in the proportion of 2:3.  So if AC = 8, DF would equal 12, and if FE = 6, CB would equal 4.

Similar triangles often show up layered, as the smaller triangle inside the larger, like this:

In a situation like this, you need the small triangle and the large triangle to share an angle, and also for the third side of each triangle to be parallel.  Only with those conditions in place can you treat these two triangles as similar.

Two triangles with congruent angles are similar, but only when all the angles and all the sides are equal are the triangles themselves considered congruent.  This is a rare occurrence, so be careful - don't assume the sides are equal in length when you have equal angles, because they most likely are not equal, but proportional to each other.

Students with Starting Score Above 700:

When dealing with similar triangles, there is also a shortcut to find the ratio of their areas and volumes.  The ratio of the areas of similar triangles is the ratio of their sides, squared.  So two similar triangles with sides in the proportion 3/4 have areas in the proportion 9/16.

            -SAT Math Strategy: Size It Up.  Remembering the Strategy to Eliminate Overly Obvious Answers, another tool you have is to make educated guesses as to the measurement of angles to help you narrow your choices.   Take a look at these "benchmark" angles:

Using your familiarity with these angles, you can Size Up other angles that you don't know the measurement for and take a guess at what those measurements might be.  This will help you narrow down your answer choices.

Test your ability to guess well by looking at these angles:

            -SAT Math Hint: When guessing at angles, draw in a 90° or 180° line to help you estimate.

Angle 1's real value is 27°.  Angle 2's real value is 146°.  Angle 3's real value is 72°.  If you were within 5 degrees on any of your guesses, that's an excellent guess.  It's rare that an answer choice would ever be any closer than 5°, so definitely use this strategy to help eliminate obviously wrong answer choices.  If you were off by more than 10 degrees, it would help to study the benchmark angles to help you make better guesses.  The ability to make an educated guess as to the measurement of an angle is very helpful on this test.