Finding the Perimeter

8cmp06te_LP4.qxd 2/9/06 9:24 AM Page 87

4.4 Finding the Perimeter

Goal Explore 4.4

• Use the properties of special right triangles to Circulate as groups explore the problem. Some solve problems may need help identifying the three 30-60-90 In this problem, students will apply what they triangles embedded in the figure. Suggest they have learned about the Pythagorean Theorem and draw the three triangles separately as shown here: the special properties of 30-60-90 triangles. C C

30؇ 60؇ Launch 4.4

Display Transparency 4.4 on the overhead. 60؇ 30؇ C AD B 8 units D

30؇ A B 8 units D 60؇ 30؇ A B Suggested Questions Ask: • Look at triangle ABC. What do you need to Students may have different strategies for know to find its perimeter? (The lengths of determining the missing measures. Some may start the sides) with triangle BCD, some with triangle ABC. • How can we find those lengths? Suggested Questions Ask: Let students offer their ideas. They may notice • How can you find the measure of angle BCD? that the length of the side opposite the 308 angle in [This is a right triangle, so the measure is triangle ABC must be half the length of the 1808 -(908 + 308) = 608.] hypotenuse but that neither of those two lengths is • How can you find the measure of angle CAD? INVESTIGATION given. Some may notice that the measure of angle (You can use triangle ABC or triangle ACD. CAB must be 608, because the sum of the measures In the latter case, you will need to find the of the other two angles in triangle ABC is 1208. measure of angle ACD first.) • The challenge for you in this problem is to Encourage groups to keep track of their reason about the relationships in 30-60-90 calculations in an orderly way so they will be triangles and the measures that are given to able to explain their reasoning to the class. find the side lengths of triangle ABC and 4 calculate the perimeter. Have the class work in groups of four on the problem.

Investigation 4 Using the Pythagorean Theorem 87 8cmp06te_LP4.qxd 2/9/06 9:24 AM Page 88

Summarize 4.4 16 units and 32 units, respectively, the length of side BC is the square root of 322 - 162, or 768. Ask one of the groups to describe how they found " the perimeter of ABC. Here is one possible C explanation: 30؇ ؇ Because the two labeled angles in triangle ABC 60 768 units (16 3) have measures 308 and 908, the measure of angle 16 units Á Á CAB must be 1808 - 1208, or 608. Therefore, angle ACD measures 1808 - 1508 = 308, and angle 60؇ 30؇ - = A B DCB measures 908 308 608. 8 units D The side opposite the 308 angle in right triangle 32 units ACD has a length of 8 units. The length of the hypotenuse, side AC, must be twice that, or The perimeter of triangle ABC is thus 16 units. 16 + 32 + 768 < 16 + 32 + 27.7 < 75.7 units. " C Move on to the rest of the questions. Some 30؇ students may recall the properties of a 60؇ 16 units 30-60-90 triangle and realize that the length of BC is 16 3. # 60؇ 30؇ Suggested Questions Once students have A B 8 units D discussed how they found the areas of the triangles, ask: Because side AB is the hypotenuse of the • What is the relationship between the areas of 30-60-90 triangle ABC and the length of the side the two smaller triangles and the area of the opposite the 30° angle is 16 units, the length of largest triangle? (The sum of the areas of the the hypotenuse, or side AB, must be twice that, two smaller triangles is equal to the area of or 32 units. the largest triangle.) We can now apply the Pythagorean Theorem to • Which triangles are similar? Why? find the missing side length of triangle ABC. For each pair of similar triangles, what is the Because one leg and the hypotenuse measure • ratio of the short leg to the long leg? The short leg to the hypotenuse?

88 Looking for Pythagoras 8cmp06te_LP4.qxd 2/9/06 9:24 AM Page 89

At a Glance 4.4 Finding the Perimeter PACING 1 day Mathematical Goal • Use the properties of special right triangles to solve problems

Launch Materials Display Transparency 4.4 on the overhead. • Transparency 4.4 Look at triangle ABC. What do you need to know to find its perimeter? • • Labsheet 4.4 How can we find those lengths? Let students offer their ideas. • The challenge for you in this problem is to reason about the relationships in 30-60-90 triangles and about the measures that are given to find the side lengths of triangle ABC and then to calculate the perimeter. Have the class work in groups of four on the problem.

Explore

Circulate as groups explore the problem. Some may need help identifying the three 30-60-90 triangles embedded in the figure. Suggest they draw the three triangles separately. • How can you find the measure of angle BCD? How can you find the measure of angle CAD? Encourage groups to keep track of their calculations in an orderly way so they will be able to explain their reasoning to the class.

Summarize Materials Ask one of the groups to describe how they found the perimeter of ABC. • Student notebooks Move on to the rest of the questions. Once students have discussed how they found the areas of the triangles, ask: • What is the relationship between the areas of the two smaller triangles and the area of the largest triangle?

Investigation 4 Using the Pythagorean Theorem 89 8cmp06te_LP4.qxd 2/9/06 9:24 AM Page 90

ACE Assignment Guide B. The area of triangle ABC is for Problem 4.4 1bh = 1 ? 16 ? 768 < 221.7 units2 or 2 2 " 1 Core 12, 35 equivalently ? 32 ? 192 < 221.7 units2. 2 " Other Extensions 53–58; unassigned choices from C. Using the Pythagorean Theorem, because earlier problems 162 - 82 = 192, the length of side CD is Adapted For suggestions about adapting ACE 192, or 8 3 units. So, the area of triangle exercises, see the CMP Special Needs Handbook. " # ACD is 1bh = 1 ? 8 ? 192 < 55.4 units2.The Connecting to Prior Units 35: Stretching and 2 2 " Shrinking length of side BD is 32 – 8 = 24 units, so the area of triangle BCD is Answers to Problem 4.4 1 ? 24 ? 192 < 166.3 units2. 2 " Alternatively, some students may argue that A. Side AC has a length of 16 units, side AB has the areas of the two smaller triangles need to a length of 32 units, and side BC has a length add to the area of the largest triangle. These of 768 , or 16 3 units. The perimeter of students will use the area formula to find the " # triangle ABC is thus area of one of the smaller triangles, and then 16 + 32 + 768 < 75.7 units. Answers will subtract from the area of the largest triangle. " to find area of the other smaller triangle. vary. See the possible explanation in the Summarize section.

90 Looking for Pythagoras