1. What is the value of x?
2. What is the value of x?
3. PQRS is a square. What is its perimeter? Area?
4. ABCD is a rectangle with length 7 and width 3. What is its perimeter? Area?
5. STUV is a square. Its perimeter is 12. What is its area?
6. DEFG is a square. Its area is 81. What is its perimeter?
7. JKLM is a rectangle. If its width is 4, and its perimeter is 20, what is its area?
8. WXYZ is a rectangle. If its length is 6 and its area is 30, what is its perimeter?
9. What is the volume of a rectangular solid with height 3, width 4, and length 2?
When You Are Done
Check your answers in Chapter 3.
Triangles—Elementary, Lower, Middle, Upper Levels Only
A triangle is a geometric figure with three sides.
Isosceles Triangles
Any triangle with two equal sides is an isosceles triangle.
If two sides of a triangle are equal, the angles opposite those sides are always equal. Said another way, the sides opposite the equal angles are also equal.
This particular isosceles triangle has two equal sides (of length 6) and therefore two equal angles (40° in this case).
If you already know that the above triangle is isosceles, then you also know that y must equal one of the other sides and n must equal one of the other angles. Since n = 65 (65° + 50° + n° = 180°), then y must equal 9, because it is opposite the other 65° angle.
Equilateral Triangles
An equilateral triangle is a triangle with three equal sides. If all the sides are equal, then all the angles must be equal. Each angle in an equilateral triangle is 60°.
Right Triangles
A right triangle is a triangle with one 90° angle.
Area
To find the area of a triangle, multiply by the length of the base by the length of the triangle’s height, or .
What is the area of a triangle with base 6 and height 3?
(A) 3 (B) 6 (C) 9 (D) 12 (E) 18
Elementary and Lower Levels
The test-writers may give you the formula for the area of a triangle, but memorizing it will still save you time!
Just put the values you are given into the formula and do the math.
That’s all there is to it!
b × h = area
( ) (6) × (3) = area 3 × 3 = 9 So, (C) is the correct answer.
Don’t Forget!
A = bh
Remember the base and the height must form a 90°-angle.
The only tricky point you may run into when finding the area of a triangle is when the triangle is not a right triangle. In this case, it becomes slightly more difficult to find the height, which is easiest to think of as the distance to the point of the triangle from the base.
Here’s an illustration to help.
First look at triangle BAC, the unshaded right triangle on the left side.
Finding its base and height is simple—they are both 3. So using our formula for the area of a triangle, we can figure out that the area of triangle BAC is 4 .
Now let’s think about triangle BCD, the shaded triangle on the right. It isn’t a right triangle, so finding the height will involve a little more thought. Remember the question, though: how far up from the base is the point of triangle BCD? Think of the shaded triangle sitting on the floor of your room. How far up would its point stick up from the floor?
Yes, 3! The height of triangle BCD is exactly the same as the height of triangle BAC. Don’t worry about drawing lines inside the shaded triangle or anything like that, just figure out how high its point is from the ground.
Okay, so just to finish up, to find the base of triangle BCD (the shaded one), use the same area formula, and just plug in 3 for the base and 3
for the height.
And once you do the math, you’ll see that the area of triangle BCD is 4 .
Not quite convinced? Let’s look at the question a little differently. The base of the entire figure (triangle DAB) is 6, and the height is 3. Using your trusty area formula, you can determine that the area of triangle DAB is 9. You know the area of the unshaded triangle is 4 , so what’s left for the shaded part? You guessed it, 4 .
Similar Triangles—Middle and Upper Levels Only
Similar triangles are triangles that have the same angles but sides of different lengths. The ratio of any two corresponding sides will be the same as the ratio of any other two corresponding sides. For example, a triangle with sides 3, 4, and 5 is similar to a triangle with sides of 6, 8, and 10, because the ratio of each of the corresponding sides (3:6, 4:8, and 5:10) can be reduced to 1:2.
One way to approach similar triangles questions that ask you for a missing side is to set up a ratio or proportion. For example, look at the question below:
What is the value of EF?
(A) 4 (B) 5 (C) 6 (D) 7 (E) 8
These triangles are similar because they have the same angles. To find side EF, you just need to set up a ratio or proportion.
Cross-multiply to get 15(EF) = 18(5).
Divide both sides by 15 to get EF = 6.
Therefore, the answer is (C), 6.
The Pythagorean Theorem—Upper Level Only
For all right triangles, a2 + b2 = c2, where a, b, and c are the lengths of the triangle’s sides.
Try It!
Test your knowledge of triangles with the problems that follow. If the question describes a figure that isn’t shown, make sure you draw the figure yourself!
Always remember that c represents the hypotenuse, the longest side of the triangle, which is always opposite the right angle.
1. What is the length of side BC?
(A) 4 (B) 5 (C) 6 (D) 7
(E) 8
Just put the values you are given into the formula and do the math, remembering that line BC is the hypotenuse:
a2 + b2 = c2 32 + 42 = c2 9 + 16 = c2 25 = c2 5 = c
So, (B) is the correct answer.