Triangles

Transcription

1 Page of Triangles Goal Find the side lengths of --0 triangles. Key Words --0 triangle isosceles triangle p. 7 leg of a right triangle p. hypotenuse p. Geo-Activity Eploring an Isosceles Right Triangle Fold a large piece of paper so the top lines up with one side. Measure the angles of the triangle formed. Measure the legs of the triangle. 4 Use the Pythagorean Theorem to predict the length of the hypotenuse. Measure the hypotenuse to verify your answer in Step 4. Student Help LOOK BACK To review the Pythagorean Theorem, see p.. A right triangle with angle measures of,, and 0 is called a --0 triangle. You can use the Pythagorean Theorem to find the length of the hypotenuse of any --0 triangle p p 6 p 4 p p 6 p 4 THEOREM Triangle Theorem Words In a --0 triangle, the length of the hypotenuse is the length of a leg times. Symbols hypotenuse legp 4 Chapter 0 Right Triangles and Trigonometry

2 Page of 6 IStudent Help I C L A S S Z O N E. C O M MORE EXAMPLES More eamples at classzone.com EXAMPLE Find the length of the hypotenuse in the --0 triangle shown at the right. Find Hypotenuse Length By the --0 Triangle Theorem, the length of the hypotenuse is the length of a leg times. hypotenuse legp p --0 Triangle Theorem Substitute. ANSWER The length of the hypotenuse is. EXAMPLE Find Leg Length Find the length of each leg in the --0 triangle shown at the right. 7 Student Help READING TIP The epression is equivalent to. By the --0 Triangle Theorem, the length of the hypotenuse is the length of a leg times. hypotenuse legp Triangle Theorem Substitute. Divide each side by. 7 Simplify. ANSWER The length of each leg is 7. Find Hypotenuse and Leg Lengths Find the value of Triangles 4

3 Page of 6 EXAMPLE Identify --0 Triangles Determine whether there is enough information to conclude that the triangle is a --0 triangle. Eplain your reasoning. By the Triangle Sum Theorem, So, 0, and 4. ANSWER Since the measure of each acute angle is, the triangle is a --0 triangle. Eample shows that whenever a right triangle has congruent acute angles, it is a --0 triangle. Student Help EXAMPLE 4 Find Leg Length LOOK BACK To review the Base Angles Theorem, see p. 8. Show that the triangle is a --0 triangle. Then find the value of. The triangle is an isosceles right triangle. By the Base Angles Theorem, its acute angles are congruent. From the result of Eample, this triangle must be a --0 triangle. You can use the --0 Triangle Theorem to find the value of. hypotenuse legp Substitute. --0 Triangle Theorem Divide each side by. Simplify.. Use a calculator to approimate. Find Leg Lengths 44 Chapter 0 Right Triangles and Trigonometry Show that the triangle is a --0 triangle. Then find the value of. Round your answer to the nearest tenth

4 Page 4 of 6 0. Eercises Guided Practice Vocabulary Check. How many congruent sides does an isosceles right triangle have?. How many congruent angles does an isosceles right triangle have? What are the measures of the three angles? Skill Check Find the value of in the --0 triangle. Write your answer in radical form Practice and Applications Etra Practice See p. 6. Finding Hypotenuse Lengths Find the length of the hypotenuse in the --0 triangle. Write your answer in radical form Finding Leg Lengths Find the length of a leg in the --0 triangle Homework Help Eample : Es. 6, 8 Eample : Es. 7 Eample : Es. 7 Eample 4: Es Triangles 4

5 Page of 6 8. Jewelry Use a calculator to find the length of the earring shown at the right. Round your answer to the nearest tenth..4 cm.4 cm You be the Judge Determine whether there is enough information to conclude that the triangle is a --0 triangle. Eplain your reasoning Finding Leg Lengths Show that the triangle is a --0 triangle. Then find the value of each variable. Round to the nearest tenth y 0 Technology In Eercises 8 0, use geometry software. Draw AB&* and construct its midpoint, C. Construct the perpendicular bisector of AB &*. Construct point D on the bisector and construct AD &* and DB &*. 4 Measure aadb. Drag point D until maadb Name three --0 triangles. Eplain how you know they are --0 triangles.. Measure AC &*, CB &*, and CD &*. What do you notice? Eplain. 0. Predict the measures of AD &* and DB &*. Check your answer by measuring the segments. A D C B. Error Analysis A student labels a --0 triangle as shown. Eplain and correct the error. 46 Chapter 0 Right Triangles and Trigonometry

6 Page 6 of 6 Quilt Design The quilt design in the photo is based on the pattern in the diagram below. Use the diagram in Eercises and. r s t u v w Wheel of Theodorus, by Diana Venters. Working from left to right, use the Pythagorean Theorem in each right triangle to find the values of r, s, t, u, v, and w.. Identify any --0 triangles in the figure. Standardized Test Practice 4. Multi-Step Problem Use the triangle shown below. a. Find the value of. Then find maa, mab, and mac. b. Find the values of b and c. c. Use the Pythagorean Theorem or the --0 Triangle Theorem to justify your answers in part (b). A b C 6 c B Mied Review Classifying Triangles Classify the triangle as acute, right, or obtuse. (Lesson 4.) Algebra Skills Simplifying Radicals Simplify the radical epression. (Lesson 0.) Writing Fractions as Decimals Write the fraction as a decimal. For repeating decimals, also round to the nearest hundredth. (Skills Review, p. 67) Triangles 47