Investigation 4.2

Investigation 4.2Investigation 4.2

AMSTIAMSTI

Searching for PythagorasSearching for Pythagoras

Problem of the DayProblem of the Day

Using the regular Using the regular hexagon, find the total hexagon, find the total number of equal number of equal triangles inside. Draw triangles inside. Draw the triangles from point the triangles from point AA. How many triangles . How many triangles did you find?did you find?

How many degrees are How many degrees are in the hexagon?in the hexagon?

How did you find this How did you find this answer?answer?

•A

Problem of the DayProblem of the Day

There are six triangles There are six triangles in a regular hexagon.in a regular hexagon.

Each triangle is 180Each triangle is 180°.°. The total number of The total number of

degrees in a regular degrees in a regular hexagon is 720°.hexagon is 720°.

You can use the You can use the formula to find the formula to find the answer!answer!

(n – 2)180°(n – 2)180°

•A

Problem 4.2 Problem 4.2 (Labsheet 4.2)(Labsheet 4.2)

Each side of Each side of equilateral triangle equilateral triangle ABCABC has a length of has a length of 2.2.

Remember, all sides Remember, all sides are equal and all are equal and all angles are 60angles are 60°.°.

A

C B

2 2

2

Labsheet 4.2Labsheet 4.2

On labsheet 4.2, find the On labsheet 4.2, find the point halfway between point halfway between vertices vertices BB and and CC. Label . Label this point this point PP..

Point P is the midpoint of Point P is the midpoint of segment segment BCBC..

Draw a segment from Draw a segment from vertex vertex AA to point to point PP..

This divides triangle This divides triangle ABCABC into two congruent into two congruent triangles.triangles.

A

BCP

Labsheet 4.2Labsheet 4.2

Cut out triangle Cut out triangle ABCABC and fold it along line and fold it along line APAP..

What do you notice What do you notice about the two new about the two new triangles?triangles?

A

BCP

Problem 4.2 (A)Problem 4.2 (A)

How does triangle How does triangle ABPABP compare with compare with triangle triangle ACPACP??

Problem 4.2 (B)Problem 4.2 (B)

Find the measure of each angle in triangle Find the measure of each angle in triangle ABPABP. Explain how you found each measure.. Explain how you found each measure.

Problem 4.2 (C)Problem 4.2 (C)

Find the length of each side of triangle Find the length of each side of triangle ABPABP. Explain how you found each . Explain how you found each length.length.

Problem 4.2 (D)Problem 4.2 (D)

Two line segments that meet at right angles are Two line segments that meet at right angles are called perpendicular line segments. Find a pair called perpendicular line segments. Find a pair of perpendicular line segments in the drawing.of perpendicular line segments in the drawing.

Paper FoldingPaper Folding

Fold the corners and draw a line. Shade in this area.

Fold the corners and draw a line. Shade in this area. Fold your

paper up the middle and draw a line.

Paper FoldingPaper Folding

You should have lines like these drawn on your envelope.

Trace the two other folds so that you have two more lines like these!

What kind of triangles do you see?

30°30°

90° 90°60° 60°

60° 60°

60°

60° 60°

60°

Problem 4.2 Follow-Up Problem 4.2 Follow-Up (1)(1)

A right triangle with a A right triangle with a 6060°° angle is sometimes angle is sometimes called a 30-60-90 called a 30-60-90 triangle. This 30-60-90 triangle. This 30-60-90 triangle has a triangle has a hypotenuse of length hypotenuse of length 6. 6.

What are the lengths of What are the lengths of the other two sides? the other two sides?

Explain how you found Explain how you found your answers.your answers.

6

30°

60°

Problem 4.2 Follow-Up Problem 4.2 Follow-Up (2)(2)

Square Square ABCDABCD has has sides of length 1. sides of length 1.

On Labsheet 4.2, On Labsheet 4.2, draw a diagonal, draw a diagonal, dividing the square dividing the square into two triangles. into two triangles.

Cut out the square Cut out the square and fold it along the and fold it along the diagonal.diagonal.

A B

D C

Problem 4.2 Follow-Up Problem 4.2 Follow-Up (2)(2)

a.a. How do the two How do the two triangles compare?triangles compare?

b.b. What are the measures What are the measures of the angles of one of of the angles of one of the triangles? Explain.the triangles? Explain.

c.c. What is the length of What is the length of the diagonal? Explain the diagonal? Explain how you found the how you found the length.length.

d.d. Suppose square Suppose square ABCDABCD had sides of length 5. had sides of length 5. How would this change How would this change your answers to parts b your answers to parts b and c?and c?

A B

D C

ACE Questions for 4.2ACE Questions for 4.2

Answer the following ACE questions on Answer the following ACE questions on page 47 - #2, 4, 7, and 10page 47 - #2, 4, 7, and 10

Investigation 4.3Investigation 4.3

AMSTIAMSTI

Searching for PythagorasSearching for Pythagoras

Investigation 4.3Investigation 4.3

Special TrianglesSpecial Triangles

For advanced students For advanced students or extra enrichmentor extra enrichment

Special TrianglesSpecial Triangles

With an equilateral With an equilateral triangle:triangle: All sides are equal All sides are equal

(shown by “a”)(shown by “a”) All angles are equal All angles are equal

(60(60°)°) The perpendicular The perpendicular

bisector is the height bisector is the height (h) and it creates two (h) and it creates two 30-60-90 triangles! 30-60-90 triangles!

4.3

Special Triangles (30-60-90)Special Triangles (30-60-90)

By using a By using a perpendicular perpendicular bisector, the shorter bisector, the shorter leg is half of the leg is half of the hypotenuse. (a/2)hypotenuse. (a/2)

Use this with the Use this with the Pythagorean Pythagorean Theorem to find Theorem to find missing lengths!missing lengths!

4.3

Special TrianglesSpecial Triangles

This right triangle This right triangle is a 45-45-90.is a 45-45-90.

The two legs are The two legs are equal (a).equal (a).

The hypotenuse The hypotenuse (h) can be found (h) can be found by using the by using the formula h/formula h/√a√a or or the Pythagorean the Pythagorean Theorem.Theorem.

4.3

Problem (pg 45)Problem (pg 45) If length If length CDCD is 8, is 8,

what is the length what is the length of of ACAC??

Now, find the Now, find the length of length of ADAD. . What length did What length did you find?you find?

What is the length What is the length of of BCBC??

What is the length What is the length of of ABAB??

8

60°30°

A

BC D

ACE Problems ACE Problems #8, 9, 11, 12#8, 9, 11, 12