Right Triangles and Pythagorean Theorem

Out of all the ACT Math topics tested, the triangle is the most commonly tested Geometric shape. The Pythagorean Theorem and the properties of right triangles are especially tested.

The Pythagorean Theorem is a formula used to find the third side of a right triangle if you know the other two sides. It only works for a right triangle, a triangle with a 90-degree angle.

The Pythagorean Theorem can be represented as

a2 + b2 = c2

in which a and b are the two shorter sides and c is the hypotenuse. Hypotenuse is the name given to the side across from the 90-degree angle. To save time, you can memorize and recognize the common Pythagorean triplets. These are common ratios between the sides that come up often in right triangles. The most common are 3:4:5 and 5:12:13. These ratios are also true for any multiples of 3:4:5 and 5:12:13 such as 6:8:10 or 10:24:26.

For example, if you are told a right triangle has a hypotenuse of 10 and one side with a length of 6, you can tell that the third side is 8. That’s because it must fit a 6:x:10 ratio, which if we divide by two becomes 3:(x/2):5. This looks like 3:4:5, so x/2 = 4, and x = 8.

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.

Example Question #1

Which of the following sets of three numbers could be the side lengths, in yards, of a right triangle containing a 45° angle?

Here, even if you forgot the 45-45-90 ratios, you can eliminate any choice that doesn’t meet the Pythagorean theorem, since the question tells you are dealing with a right triangle. So, (A) and (B) are eliminated. To choose between (C), (D), and (E), notice that a right triangle with one 45 degree angle must have another 45 degree angle. 180 - 90 - 45 = 45. This means that the triangle is isosceles, and the correct answer choice must contain two equal values. The correct answer is (C).

Example Question #2

A right triangle with one length of 60 degrees has two leg lengths of x of and 2√3. If side 2√3 is across from the 60-degree angle, what is the measurement of x?

  • A) 2
  • B) 2√3
  • C) 4
  • D) 4√2
  • E) 4√3

It is a good example of how drawing the triangle can be vital to getting the correct answer on a right triangle question. A 30-60-90 triangle has a ratio of x: x√3: 2x. If 2√3 is across from the 60-degree angle, then x = 2. The correct answer is (A).

Example Question #3

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, you can plug x in to get: 2: 2√3 : 2(2) or 2: 2√3 : 4. The answer is (E).

Example Question #4

Which of the following sets of three numbers could be the side lengths, in yards, of a right triangle containing a 45° angle?

 

A fractional exponent is just another way of expressing a root. 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. 

Example Question #5

What is the area of the trapezoid in the figure provided?

  • (A) 210
  • (B) 360
  • (C) 440
  • (D) 570
  • (E) 620

The trapezoid consists of two smaller shapes: a triangle and a rectangle. Draw a line from point A across this figure to form the two shapes. Find the area of each shape and then add them together to find the area of the trapezoid.

Area of rectangle = length x width. Therefore, the area of the rectangle is 18×20 = 360.

Area of the triangle = (1/2)(20)(21) = 210. The total area of the trapezoid = 360+210 = 570, or (D).