Scalene triangles

In addition to having three unequal sides, scalene triangles have three unequal angles. The shortest side is across from the smallest angle, the medium side is across from the medium angle, and the longest side is across from the largest angle.

Isosceles triangles

An isosceles triangle has two (or three) equal sides and two (or three) equal angles. The equal sides are called legs, and the third side is the base. The two angles touching the base (which are congruent) are called base angles. The angle between the two legs is called the vertex angle. See Figure 4-1.

Equilateral triangles

An equilateral triangle has three equal sides and three equal angles (which are each ). Its equal angles make it equiangular as well as equilateral. Note that an equilateral triangle is also isosceles.

A triangle’s altitude or height

Altitude (of a triangle): A segment from a vertex of a triangle to the opposite side (or to the extension of the opposite side if necessary) that’s perpendicular to the opposite side. The opposite side is called the base.

Imagine that you have a cardboard triangle standing straight up on a table. The altitude of the triangle tells you exactly what you’d expect — the triangle’s height (h) measured from its peak straight down to the table. This height goes down to the base of the triangle that’s flat on the table. Figure 4-4 shows you an example of an altitude.

Every triangle has three altitudes, one for each side. Figure 4-5 shows the same triangle from Figure 4-4 standing up on a table in the other two possible positions: with as the base and with as the base.

You can use any side of a triangle as a base, regardless of whether that side is on the bottom. Figure 4-6 shows again with all three of its altitudes.

Here’s the lowdown on the location of a triangle’s altitudes:

• Acute triangle: All three altitudes are inside the triangle.
• Right triangle: One altitude is inside the triangle, and the other two altitudes are the legs of the triangle (remember this when figuring the area of a right triangle).
• Obtuse triangle: One altitude is inside the triangle, and two altitudes are outside the triangle.

Determining a triangle’s area

Triangle area formula:

Assume for the sake of argument that you have trouble remembering this formula. Well, you won’t forget it if you focus on why it’s true — which brings me to one of the most important tips in this book.

Whenever possible, don’t just memorize math concepts, formulas, and so on by rote. Try to understand why they’re true. When you grasp the whys underlying the ideas, you remember them better and develop a deeper appreciation of the interconnections among mathematical ideas. That appreciation makes you a more successful math student.

So why does the area of a triangle equal ?

Because the area of a rectangle is , and a triangle is half of a rectangle. Check out Figure 4-7, which shows two triangles inscribed in rectangles HALF and PINT.

It should be really obvious that has half the area of rectangle HALF. And it shouldn’t exactly give you a brain hemorrhage to see that also has half the area of the rectangle around it. (Triangle PXZ is half of rectangle PIXZ, and is half of rectangle ZXNT.) Because every possible triangle fits in some rectangle just like fits in rectangle PINT (you just have to put the triangle’s longest side on the bottom), every triangle is half a rectangle.

Now for a problem: What’s the length of altitude in in Figure 4-8?

The trick here is to note that because is a right triangle, legs and are also altitudes. So you can use either one as the altitude, and then the other leg automatically becomes the base. Plug their lengths into the formula to determine the triangle’s area:

Now you can use the area formula again, using this area of 150, base , and altitude :

The Fab Four triangles

The first four Pythagorean triple triangles are the favorites of geometry problem makers. These triangles, especially the first and second on the list, pop up all over in geometry books.

Here are the first four Pythagorean triple triangles:

• The 3-4-5 triangle
• The 5-12-13 triangle
• The 7-24-25 triangle
• The 8-15-17 triangle

Families of Pythagorean triple triangles

Each irreducible Pythagorean triple triangle such as the 5-12-13 triangle is the matriarch of a family with an infinite number of children. The 3 : 4 : 5 family (note the colons), for example, consists of the 3-4-5 triangle and all her offspring. Offspring are created by blowing up or shrinking the 3-4-5 triangle: They include the triangle, the 6-8-10 triangle, the 21-28-35 triangle (3-4-5 times 7), and their eccentric siblings such as the triangle and the triangle. Within any of the triangle families (like the 3 : 4 : 5 family), all the triangles have the same shape.

When you know two of the three sides of a right triangle, you can, of course, compute the third side with the Pythagorean Theorem. But if the triangle happens to be a member of one of the Fab Four families, you can use a shortcut. All you need to do is figure out the blow-up or shrink factor that converts the main Fab Four triangle into the given triangle and use that factor to compute the third side of the given triangle.

No-brainer cases

You can often just see that you have one of the Fab Four families and figure out the blow-up or shrink factor in your head. Check out Figure 4-11.

In Figure 4-11a, the digits 8 and 17 in the 0.08 and 0.17 should give you a big hint that this triangle is a member of the 8 : 15 : 17 family. Because 8 divided by 100 is 0.08 and 17 divided by 100 is 0.17, this triangle is an 8-15-17 triangle shrunk down 100 times. Side j is thus 15 divided by 100, or 0.15. This shortcut is way easier than using the Pythagorean Theorem.

Likewise, the digits 3 and 4 should make it a dead giveaway that the triangle in Figure 4-11b is a member of the 3 : 4 : 5 family. Because is times 3 and is times 4, you can see that this triangle is a 3-4-5 triangle blown up by a factor of . Thus, side r is simply times 5, or .

Make sure the sides of the given triangle match up correctly with the sides of the Fab Four triangle family you’re using. In a 3 : 4 : 5 triangle, for example, the legs must be the 3 and the 4, and the hypotenuse must be the 5. So a triangle with legs of 30 and 50 (despite the 3 and the 5) is not in the 3 : 4 : 5 family because the 50 (the 5) is one of the legs instead of the hypotenuse.

The step-by-step triple triangle method

If you can’t immediately see what Fab Four family a triangle belongs to, you can always use the following step-by-step method to pick the family and find the missing side. Don’t be put off by the length of the method; it’s easier to do than to explain. Check out the triangle in Figure 4-12.

1. Take the two known sides and make a ratio in fraction form of the smaller to the larger side.

   Take the and the 6 and make the ratio of .

2. Reduce this ratio to whole numbers in lowest terms.

   If you multiply the top and bottom of by 5, you get ; that reduces to . (With many calculators, this is a snap because they have a function that reduces fractions to lowest terms.)

3. Look at the fraction from Step 2 to spot the particular triangle family.

   The numbers 4 and 5 are part of the 3-4-5 triangle, so you’re dealing with the 3 : 4 : 5 family.

4. Divide the length of a side from the given triangle by the corresponding number from the family ratio to get your multiplier (which tells you how much the basic triangle has been blown-up or shrunk).

   Use the length of the hypotenuse from the given triangle (because working with a whole number is easier) and divide it by the 5 from the 3 : 4 : 5 ratio. You should get or 1.2 for your multiplier.

5. Multiply the third family number (the number you don’t see in the reduced fraction in Step 2) by the result from Step 4 to find the missing side of your triangle.

   Three times is . That’s the length of side p; and that’s a wrap.

You may be wondering why you should go through all this trouble when you could just use the Pythagorean Theorem. Good point. The Pythagorean Theorem is easier for some triangles (especially if you’re allowed to use your calculator). But — take my word for it — this triple triangle technique can come in handy. Take your pick.

The triangle

The triangle (or isosceles right triangle): A triangle with angles of , , and and sides in the ratio of . Note that it’s the shape of half a square, cut along the square’s diagonal, and that it’s also an isosceles triangle. See Figure 4-13.

Try a couple of problems. Find the lengths of the unknown sides in triangles BAT and BOY shown in Figure 4-14.

You can solve triangle problems in two ways: the formal book method and the street-smart method. Try ’em both and take your pick. The formal method uses the ratio of the sides from Figure 4-13.

For , because one of the legs is 8, the x in the ratio is 8. Plugging 8 into the three x’s gives you

And for , the hypotenuse is 10, so you set the from the ratio equal to 10 and solve for x:

That does it:

Now for the street-smart method (this uses the same math as the formal method, but it involves fewer steps): Remember the triangle as the “ triangle.” Using that tidbit, do one of the following:

• If you know a leg and want to compute the hypotenuse (a longer thing), you multiply by . In Figure 4-14, one of the legs in is 8, so you multiply that by to get the longer hypotenuse — .
• If you know the hypotenuse and want to compute the length of a leg (a shorter thing), you divide by . In Figure 4-14, the hypotenuse in is 10, so you divide that by to get the shorter legs; they’re each or .

The triangle

The triangle: A triangle with angles of , , and and sides in the ratio of . Note that it’s the shape of half an equilateral triangle, cut straight down the middle along its altitude. Check out Figure 4-15.

Here are a couple of problems. Find the lengths of the unknown sides in and in Figure 4-16.

You can solve triangles with the textbook method or the street-smart method. The textbook method begins with the ratio of the sides from Figure 4-15:

In , the hypotenuse is 10, so you set 2x equal to 10 and solve for x, getting . Now just plug 5 in for the x’s, and you have :

In , the long leg is 9, so set equal to 9 and solve:

Plug in the value of x, and you’re done:

Here’s the street-smart method. Think of the triangle as the “ triangle.” Using that fact, do the following:

• The relationship between the short leg and the hypotenuse is a no-brainer: The hypotenuse is twice as long as the short leg. So if you know one of them, you can get the other in your head. The method mainly concerns the connection between the short and long legs.
• If you know the short leg and want to compute the long leg (a longer thing), you multiply by . If you know the long leg and want to compute the short leg (a shorter thing), you divide by .

Try the street-smart method with the triangles in Figure 4-16. The hypotenuse in is 10, so first you cut that in half to get the length of the short leg, which is thus 5. Then to get the longer leg, you multiply that by , which gives you . In , the long leg is 9, so to get the shorter leg, you divide that by , which gives you , or . The hypotenuse is twice that, .

With the triangle (and also with the triangle), there will almost always be one or two sides whose lengths contain a square root symbol (in unusual cases, all three sides could contain a radical symbol). But it’s impossible to have no square roots — which brings me to the following warning.

Because at least one side of a triangle must contain a square root, a triangle cannot belong to any of the Pythagorean triple triangle families. So don’t make the mistake of thinking that a triangle is in, say, the 8 : 15 : 17 family or that any triangle that is in one of the Pythagorean triple triangle families is also a triangle. There’s no overlap between the triangle (or the triangle) and any of the Pythagorean triple triangles and their families.