PROPORTION PROBLEMS

GEOMETRY (advanced)

Concept: A ratio is a comparison of two numbers. It can be written in several ways. For example, when comparing the numbers 1 and 2, we can express the comparison as 1 to 2, or 1 : 2 or ½.

Always write the terms of the ratio in the same order as the problem compares them.

Just as we can reduce fractions to lowest terms, we can simplify ratios to their lowest terms, although a ratio with a denominator of 1 must still be expressed as a fraction, and ratios that appear to be improper fractions must be left as improper fractions.

Many ratio problems require more than one step.

You may not be given both of the numbers that you need to write a ratio. In this case, you may need to solve another equation to find the missing number.

Finally, when two ratios are written as equal ratios, the equation is called a proportion. For example, is a proportion. In many proportion problems, one of the four terms is missing.

The proportion can be solved using the Cross-product Rule.

Cross-product Rule
To find the missing number in a proportion, cross multiply, and divide the product by the third number. For example, in the proportion , first multiply . Then divide . In this problem .

Steps for Solving Geometry Proportion Problems

Step 1 | Form a proportion using two ratios. Use X for the unknown number.
Step 2 | Apply the Cross-product Rule to solve for X.

Example 1: Given that Rectangles A and B are similar, and leg “d” is 78, leg “c” is 62, and the Area of B is 1209. What is the length of “g”? Round your answer to the nearest hundredth.

Step 1 | Form a proportion using two ratios. Use X for the unknown number. |
Step a | Form the first ratio using the areas of rectangles A and B. |
| You were given the area of rectangle B in the problem. Determine the area of rectangle A by multiplying side “c” by side “d”. |
| Since the measure of the areas are squared (i.e. 12092 and 48362), we must find the square root of that ratio, , before we set up our proportion. |
Step b | Form the second ratio using the measurement of side c in rectangle A and X, which represents side “g” in rectangle B. |
| Form a proportion using the square root of the ratio of the areas for the first ratio, , and for the second ratio use the measurement of side “c” in rectangle A and X, which represents side “g” in rectangle B, which would be . |
Step 2 | Apply the Cross-product Rule to solve for X. First, multiply 1 by 62. Then, divide 62 by 2. Finally, X equals 31. | X = 31
Side “g” in Rectangle B measures 31.

Example 2: Triangle A is similar to Triangle B in the ratio of 2 : 1 and “b” = 70 and “d” = 28, what is the Area of B? Round each intermediate answer (before using) and the final answer to the nearest hundredth.

Step 1 | Form a proportion using two ratios. Use X for the unknown number. |
Step a | We know that the ratio of the two triangles is 2 : 1. That will form the first ratio in our proportion. |
Step b | We know that the measurement of side “d” in Triangle B is 28. We will use X to represent the measurement of side “a” in Triangle A. This will form the second ratio in our proportion. |
| Form a proportion using the ratio of the two triangles for the first ratio, , and the ratio of side “d”, 28, and side “a”, X, for the second ratio, . | =
Step 2 | Apply the Cross-product Rule to solve for X. First, multiply 2 by 28. Then, divide 56 by 1. Finally, side “a” equals 56. | X = 56
We now have the necessary information to solve the problem. To find the area of Triangle A, we needed the measurements of both side “a” and side “b”. We have both now. Side “a” equals 56, and side “b” equals 70. The formula to find the area of a triangle is: Area= ½ bh.
The Area of Triangle A is 1960.

Example 3: Given Triangle A and B are similar and Triangle A’s area is: 7424 and Triangle B’s area is: 464 and Side “b” has length 128, what is the length of “f”? Round answer to the nearest hundredth.

Step 1 | Form a proportion using two ratios. Use X for the unknown number. |
Step a | Form a ratio with the areas of Triangle A and Triangle B. |
| Since the measure of the areas are squared (i.e. 4642 and 74242), we must find the square root of that ratio, , before we set up our proportion . |
Step b | Form the second ratio using the measurement of side “b” in Triangle A and X, which represents side “e” in Triangle B. |
| Form a proportion using the square root of the ratio of the areas for the first ratio, , and for the second ratio use the measurement of side “b” in Triangle A and X, which represents side “e” in Triangle B, . | =
Step 2 | Apply the Cross-product Rule to solve for X. First, multiply 21.541 by 128. Then, divide 2757.25 by 86.163. Finally, side “e” equals 32. | X = 32
We know the area of Triangle B, 464, and the length of side “e”, 32. Using the formula for a triangle, Area= ½ bh, we can now determine the length of side “d”.
Area = ½ bh The length of side “d” is 29.
We now know the length of both sides “e” (32) and “d” (29). Using the Pythagorean Theorem, , we can now find the length of side “f”.
The length of side “f” is 43.19.

Example 4: Given Triangle A and B are similar and Triangle A’s area is: 2656.25 and Triangle B’s area is: 425 and side “a” has length 62.5, what is the length of “f”? Round each intermediate answer (before using) and the final answer to the nearest hundredth.

Step 1 | Form a proportion using two ratios. |
Step a | Form a ratio with the sides of the Triangle A and carry that ratio over to Triangle B. From the information given we know the following: Area of Triangle A: 2656.25 Side a: 62.5 Armed with this information we can then solve for side “b” Next we determine the ratio of side a to side b And since the problem states that Triangle A is similar to Triangle B then the ratio of the legs should be the same. | Area A = ½ ab
Step b | Derive an equation from the ratio that we just obtained for Triangle B. | Equation 1: d = 0.735e
| For this problem, we see that we have two unknowns “d” and “e”. To solve for these unknowns we need to have two equations. Once we solve for “d” and “e” then we can use these answers to solve for “f”. |
Step c | We can use the area of a triangle formula to set up our second equation: From the information given we know the following: Area of Triangle B: 425 This then produces our second equation Substituting Equation 1 into Equation 2 we can then solve for “e”. d = 0.735e de = 850 Substituting this value “e” back into either Equation 1 or Equation 2 we can then solve for “d”. In this case I’ll use Equation 1. d = 0.735e | Equation 2: Area B = ½ de 425 = ½ de de = 850 (0.735e)e = 850 0.735e² = 850 e² = 850 ÷ 0.735 e² = 1156.46 e = 34 d = 0.735(34) d= 25 Solve for “f” by using the Pythagoras Theorem. f² = d² + e² f² = (25)² + (34)² f² = 625 + 1156 = 1781 f = 42.2
Step 2 | Apply the Cross-product Rule to solve for X. | Since this problem does not ask us to determine the ratio this step does not apply to this problem.
We now know the area of Triangle B, 425, and the length of it’s sides. Side “e”, 34, and side “d”, 25. Using the formula for Pythagoras Theorem, f² = d² + e², we can now solve for the length of side “f”.
f² = d² + e² f² = (25)² + (34)² f² = 625 + 1156 = 1781 f = 42.2
The length of side “f” is 42.2.