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Test PrepACTMathTriangles

# ACT Math Skill Review: Triangles

Unlike the SAT Math sections, the ACT Math test does not include a helpful reference box with information and formulas such as you’ll find within this skill review. But, just knowing the information isn’t enough anyway! You’ll also need to know when (and how) to apply it.

#### Basic Triangle Facts

 A triangle is a three-sided figure whose angles always add up to 180°. Thus, ∠x +∠y + ∠z = 180°. The largest angle of a triangle is opposite its longest side. On the SAT, the triangle to the left would be referred to as triangle abc or ∆abc.  To find the perimeter of the triangle, add up all the sides (a + b + c). To find the area of any triangle, use the formula, (1⁄2)bh. The height must be perpendicular to the base. Two triangles are similar if their angles have the same measure. The triangles therefore have the same shape, and their sides will be in proportion. Triangles are said to be congruent if they have the same shape and size.

#### Equilateral Triangles

 An equilateral triangle has three equal sides. Thus: a = b = c. An equilateral triangle has three angles are also equal. Thus, ∠x = ∠y = ∠z. Since the angles of a triangle always add up to 180°, each angle of an equilateral triangle must also equal 60°. Thus, ∠x = ∠y = ∠z = 60°.

#### Isosceles Triangles

 An isosceles triangle is a triangle with two sides of equal length. Thus, a = b. The angles opposite the equal sides are also equal. Thus ∠x = ∠y. You can have an Isosceles Right Triangle (see below).

#### Right Triangles

 A right triangle is a triangle with a right angle. In the example to the left, this is ∠y. The other two angles are, by definition, complementary angles (∠x + ∠y = 90). The longest side of a right triangle (the one opposite the 90˚ angle) is called the hypotenuse (side c in the figure to the left). The other sides (a and b) are often referred to as legs. The sides of a right triangle always exist in a particular proportion to each other. This is called the Pythagorean Theorem, which states: a2 + b2 = c2.  The hypotenuse is always c.

#### Special Facts about Right Triangles

• These triangles are the ones most likely to appear on the SAT.
• If you know the lengths of any two sides of a right triangle, you can find the length of the third side using the Pythagorean Theorem.
• The hypotenuse is always the longest side of a right triangle.

#### Special Right Triangles

There are two so-called “Special Right Triangles” that frequently appear on the SAT. We’ll take a look at these in greater detail:

 The lengths of a 30˚-60˚-90˚ triangle are in the ratio of 1 : √3 : 2, as shown in the figure to the left. A 45˚-45˚-90˚ triangle is an isosceles triangle containing a right angle. The two legs of the triangle (opposite the 45˚ angles) are equal.  The sides of the triangle are in the ratio of 1:1:√2, as shown in the figure to the left.

#### Examples

1. The correct answer is E. The fact that there are two overlapping triangles here makes this problem look a lot more daunting than it is. We really are concerned only with triangle ACD, which we know is a 30-60-90 triangle. Hence, this makes it one of our special right triangles. How do we know this? We know that one angle equals 30 and one angle equals 90 (a right angle). Since all the angles in a triangle must add up to 180, the missing angle (∠ADC) must equal 60. Triangle BCD only comes into play because the question stem tells us that AC = BD and BD = 10. Therefore, AC = 10. With this figure in hand, we use what we know of 30-60-90 triangles and plug in 10 = x√3 and solve for x. Thus, x =  10/√3 and the length of AC will be equal to 2x, or 20⁄√3.

2. The correct answer is F. Remember, the ratios for the lengths of the sides of 45-45-90 triangles (and 30-60-90 triangles) appear at the top of every math section. If you are still having trouble, consider that  √2 only appears in 45-45-90 triangles (30-60-90 triangles have a √3 in them). On that basis alone, if you are guessing, you could pretty safely eliminate choices G, H and K. Another tip would be to draw a triangle and write in as much information as you know. This often helps clarify the information you have. In this case, your drawing might look something like this:

Since we know the relationship between the lengths of the sides in a 45-45-90 triangle, we can use them to solve our problem. 8 = x√2. Thus, x = 8⁄√2 = 4√2.

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