ROMEO AND ROMEO AND JULIETJULIET

AND THE LADDERAND THE LADDER

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Romeo

Juliet

IntroductionIntroduction Objectives:Objectives: Recall the properties of an isosceles right Recall the properties of an isosceles right

triangle.triangle. Find the hypotenuse of a 45Find the hypotenuse of a 45°-45°-90° °-45°-90°

triangle using the Pythagorean Theorem.triangle using the Pythagorean Theorem. Apply the special triangles to find the Apply the special triangles to find the

missing sides of a 45°-45°-90° triangle.missing sides of a 45°-45°-90° triangle. Target group:Target group: Fourth Year Students Fourth Year Students Duration/Mode:Duration/Mode: 30minutes/student-centred 30minutes/student-centred Instructions:Instructions: click on the icons in the click on the icons in the

navigation bar below to move from one section navigation bar below to move from one section to anotherto another

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Romeo and Juliet Romeo and Juliet were were

very much in love with very much in love with eacheach

other. But their parents hadother. But their parents had disapproved strongly of their disapproved strongly of their

relationship because their clans are relationship because their clans are the best of enemies. the best of enemies.

Romeo wanted very much to see Romeo wanted very much to see the love of his life Juliet. So , he the love of his life Juliet. So , he

visited Juliet’s place and thought of visited Juliet’s place and thought of ways on how he could see Juliet ways on how he could see Juliet

secretlysecretly..quit

He saw that there He saw that there

was a balcony near her was a balcony near her

room . The height of the room . The height of the balcony was 5 meters , and it was balcony was 5 meters , and it was perpendicular to the ground. He also perpendicular to the ground. He also knew that it will be easier to find the knew that it will be easier to find the length of the ladder if he will place it length of the ladder if he will place it at 45at 45° angle ° angle from the ground.from the ground.

Why?Why?

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((Choose the best Choose the best reason.)reason.)

The triangle formed is The triangle formed is equiangular so the acute angles are equiangular so the acute angles are equal.equal.

The triangle formed is an The triangle formed is an isosceles right so the measure of the two isosceles right so the measure of the two legs are equal and both acute angles legs are equal and both acute angles have measures of 45have measures of 45°°. .

The triangle formed is a right The triangle formed is a right triangle with acute angles that triangle with acute angles that measures 30measures 30°° and 60 and 60°°..

Very Good !

Very Good! Very Good!

Very Good!

Please try Again!

45°

5m

45°

45°

5m

So Romeo started to make the ladder that will be his easiest way to reach Juliet.

ladder

The balcony is 5 The balcony is 5 meters high and meters high and perpendicular to the perpendicular to the ground. If the ladder ground. If the ladder will be placed 5 will be placed 5 meters from the base meters from the base of the balcony, and of the balcony, and form a 45° angle, form a 45° angle, how long will the how long will the ladder be? (Use the ladder be? (Use the Pythagorean Pythagorean Theorem)Theorem)

balcony

wall

ground

ladder

5

5

45°

90°45°

Can you guide him to find the length of the Can you guide him to find the length of the ladder?ladder?

a² + b² = c²

How do you solve for the How do you solve for the length of the ladder?length of the ladder?

1. What is the length of a leg or 1. What is the length of a leg or a?a?

2. What is the length of the 2. What is the length of the other leg or b?other leg or b?

3. Use the Pythagorean Theorem 3. Use the Pythagorean Theorem to solve for the hypotenuse or c.to solve for the hypotenuse or c.

a² + b² = c²a² + b² = c² ((Click on the icon to do your Click on the icon to do your work.)work.) answer

The length of the ladder is The length of the ladder is 5√2 meters5√2 meters..

( You can tell Romeo to start ( You can tell Romeo to start making the ladder)making the ladder)

Let us solve for c againLet us solve for c again. .

1. a= 4, b =41. a= 4, b =4 4² + 4² = c² so 16 + 16 = 4² + 4² = c² so 16 + 16 =

c². Therefore c². Therefore 32 = c², so 4√2 = c.32 = c², so 4√2 = c. 2. a = 10, b =. 102. a = 10, b =. 10 10² + 10 ² = c² so 100 + 10² + 10 ² = c² so 100 +

100 = c². Therefore100 = c². Therefore 200 = c² so 10√2 = c. 200 = c² so 10√2 = c.

What conclusion can What conclusion can you derive from the different you derive from the different examples about the length of examples about the length of the hypotenuse given a 45°-the hypotenuse given a 45°-

45°- 90° triangle?45°- 90° triangle?

The length of the hypotenuse of The length of the hypotenuse of a 45°- 45°- 90° triangle is the length a 45°- 45°- 90° triangle is the length of a leg times √3.of a leg times √3.

The length of the hypotenuse The length of the hypotenuse of a 45°- 45°- 90° triangle is one-of a 45°- 45°- 90° triangle is one-half of the length of a leg.half of the length of a leg.

The length of the hypotenuse of a length of the hypotenuse of a 45°-45°-90 ° triangle is the length of a 45°-45°-90 ° triangle is the length of a leg leg times √2. √2.

You got it

Please try again

Complete the tableComplete the table

aa bb cc

22

1212

7√27√2

1010

table

Practice Exercises

Excellent! If you got Excellent! If you got a perfect score.a perfect score.

Try some more exercises if you got a score of Try some more exercises if you got a score of 6. 6.

Go back to the examples if you got a Go back to the examples if you got a score of 4 and below.score of 4 and below.

For more readings on 45For more readings on 45°°-45-45°-°-9090°° triangle please be triangle please be encouraged to look atencouraged to look at

http://en.wikipedia.org/wiki/Special_rihttp://en.wikipedia.org/wiki/Special_right_trianglesght_triangles

http://www.onlinemathlearning.com/shttp://www.onlinemathlearning.com/special-right-triangles.htmlpecial-right-triangles.html

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EvaluationEvaluation• Send your Send your

answer to answer to sofievelasco52@yasofievelasco52@yahoo. com.hoo. com.

• Complete the Complete the table:table:

aa bb cc

33

1/21/2

3/43/4

4√24√2

66

Your assignment will be sent to you once you have send the correct answers.

CreditsCredits Prepared by: Prepared by:

Mrs. Sofia S. VelascoMrs. Sofia S. Velasco Mathematics DepartmentMathematics Department Quezon City High SchoolQuezon City High School September 18, 2008September 18, 2008

Special Thanks to: Special Thanks to: Mr. Philip WongMr. Philip Wong