ACT Math: Properties of Triangles
If geometry questions are giving you a little trouble, it’s best to start with the basics. In order to improve your score on the ACT Math test, you just need to know your basic operations and a few key concepts. Triangles is the most important geometric concept, because almost every single geometry question has something to do with triangles. Once you get more comfortable with triangles, and learn them inside and out, they will become your favorite shape.
Here are just a couple of helpful facts about triangles that will prepare you for a good portion of the questions on the ACT .
1. Every single triangle contains three angles. These three angles always add up to 180º. This is true for right triangles, equilateral triangles, isosceles triangles, acute triangles, obtuse triangles and any other kind you can think of.
2. The sum of two sides of a triangle will ALWAYS be greater than the third side. Let’s say that you have a triangle and they only give you two of the three sides. You are given 4 and 4. We know that the remaining side MUST be less than 8. You can look at your answer choices and cross out anything that is 8 or greater.
3. The difference of two sides of a triangle will ALWAYS be less than the third side. This is an extension of the second fact. Using the same example, we are given a triangle with sides 4 and 4. The difference being 4-4 = 0. We know that the third side is greater than 0 and less than 8. Knowing this, we can head towards the correct answer.
4. There are several angle-side relationships that should be memorized.
– If two sides of a triangle are equal, then their opposite angles will be equal as well.
– The shortest side of a triangle is opposite the smallest angle.
– The longest side of a triangle is opposite the largest angle.
ACT Triangles: The Pythagorean Theorem
Right triangles and the Pythagorean Theorem are one of the most-tested ACT concepts. Here’s a online studying guide to getting better scores for these Geometry questions. The Pythagorean Theorem allows us to find the third side of any right triangle (a triangle is “right” when it contains a 90 degree angle) when we know the other two sides.
The Pythagorean Theorem states that a2 + b2 = c2. a and b are the two shorter sides and c is always the longest side (the side across from the 90 degree angle). The longest side in a right triangle is called the hypotenuse.
In this triangle, the “?” side is the hypotenuse. Let’s plug the values for the other two sides into the Pythagorean Theorem to solve:
a2 + b2 = c2
82 + 52 = c2
64 + 25 = c2
89 = c2
To remove the exponent (2), we must take the square root (√) of both sides.
√89 = c
To save time on the ACT (and let’s face it, with 60 questions to answer in 60 minutes, we could all use more time!) memorize the common Pythagorean triplets.
Rather than use the Pythagorean theorem to find the third side of a right triangle every single time, you may notice that you often encounter right triangles with the ratios of 3:4:5 and 5:12:13. These ratios will also be true for any multiples of 3:4:5 and 5:12:13 such as 6:8:10 or 10:24:26.
For example, in this triangle we know the third side must be 5, even without using the Pythagorean Theorem because we know 5:12:13 is a common triplet. Be cautious, however, the 13 must always be across from the 90 degree angle.
There are also two right triangles that are very important to know called the special right triangles. These are so called because the ratio of their sides never changes. The first is a 30-60-90 triangle. Its sides will always be in a ratio of x: x√3 : 2x.
The other special triangle is the 45-45-90 triangle. Its sides will always be in a ratio of x: x: x√2.
It’s important to remember that for the 30-60-90 triangle, the hypotenuse is the side that has the ratio of 2x. Don’t confuse it with the 45-45-90 ratio, and think that the x√3 should be there!
If triangle ABC is a 30°-60°-90° right triangle, which of the following sets could represent triangle ABC’s side lengths?
A 2, 2, 2
B 2, 2, 2√2
C 2, 2√2, 2√2
D 2, 2√2, 2√3
E 2, 2√3, 4
For each answer choice x = 2, so knowing that the ratio of a 30-60-90 is x: x√3 : 2x, we can plug x in to get: 2: 2√3 : 2(2) or 2: 2√3 : 4. The answer is E.
Now let’s look at an example with the 45-45-90 triangle:
Which of the following sets of three numbers could be the side lengths, in yards, of a right triangle containing a 45° angle?
A 1, 1, 1
B 1, 21/2, 21/2
C 2, 2, 2(21/2)
D 1, 21/2, 31/2
E 1, 31/2, 2
It’s important to recognize that a fractional exponent is just another way of expressing a root. An exponent of ½ is equal to the square root symbol, 21/2 = √2.
We know the ratio for a 45-45-90 is x: x: x√2, which means two of the sides must be equal. That eliminates D and E. Out of the remaining choices, only C correctly expresses the ratio.
Sometimes the ACT will disguise Geometry questions in word problems. This question is a perfect example:
If a public restroom is located 5 blocks south and 3 blocks east from point H, and Joe’s apartment is 2 blocks north and 6 blocks west of point H, how far is Joe’s apartment from the public restroom, to the nearest integer?
Faced with a word problem such as this, the first thing to do is draw it on your scratch paper.
Carefully following the directions in the questions, we started at point H and went down 5 and to the right 3 to get to the restroom. Then we went 2 up and 6 to the left to get to Joe’s apartment. We want to find the distance between Joe’s and the Restroom, which is represented by the blue line. Can we draw a triangle so that the blue line is the hypotenuse?
The horizontal distance from Joe’s to the Restroom is 6 + 3, so one leg of the triangle is 9. The vertical distance from Joe’s to the Restroom is 2 + 5, so the other leg of the triangle is 7. Now that we know two legs of a right triangle, we can solve. Sadly the 7:9:x ratio is not one of our Pythagorean triplets, so we’ll have to use Pythagorean theorem to solve.
a2 + b2 = c2
72 + 92 = c2
49 + 81 = c2
130 = c2
11.4 ≈ c