3.3 How Can I Find the Height? Pg. 9 Heights and Areas of Triangles

3.3 – How Can I Find the Height?__________ Heights and Area of Triangles What if you want the area of only half a rectangle? Today you will discover how to find the area of a triangle. Then you will find a shortcut to find either the hypotenuse or leg of a right triangle.

3.16 –AREA OF SHADED PART Find the area of the shaded figure. Be ready to share any observations.

html?movie=/hotmath_help/gizmos/triangleA rea.swf Area of Triangle

b h b h

3.17 –AREA OF A TRIANGLE How do you know which dimensions to use when finding the area of a triangle? a. Find the area of each shaded figure. Draw any lines on the paper that will help. Turning the triangles may help you discover a way to find their areas.

A = 1212 (21)(12) A =126 un 2

A = 1212 (6)(5) A =15 un 2

7 8 A = 1212 (7)(8) A =28 un 2

3.18 – WORKING BACKWARDS Find the missing length of the figure, given the area. Don’t forget units.

42 = 1212 (14) (h) 42 =7h h = 6 cm

216 = 1212 (24) (h) 216 =12h h = 18 in

925 =(37)(x) x = 25 ft

3.19 –SIDE OF A SQUARE What do you notice about the square at right? a. Eunice does not know how to solve for x. Explain to her how to find the missing dimension. = 10

b. What if the area of the shape above is instead 66ft 2 ? What would x be in that case? 66

3.20 –LENGTH OF THE LONG SIDE OF A TRIANGLE While Alexandria was doodling on graph paper, she made the design at right. She started with the shaded right triangle. She then rotated it 90° clockwise and translated the result so that the right angle of the image was at B. She continued this pattern until she completed the square.

a. What is the area of the large outer square? 100 un 2

b. What is the area of each triangle? 10.5 un 2

c. What is the area of the inner square? – un 2 58

d. What's the length of the longest side of the shaded triangle?

3.21 – ANOTHER WAY Robert complained that while the method from the previous problem works, it seems like too much work! He decides to use the area of the triangles, the inner square, and the large outer square to look for a short cut.

a. Find the area of the large square around the entire shape. (a+b)(a+b) a 2 + 2ab + b 2

b. Find the TOTAL area of the four triangles. Then find the area of the inner square. inner = c 2 4(½bh) = 2ab

c. Since the large square has the same area as the 4 triangles and inner square, set them equal to each other and reduce. a 2 + 2ab + b 2 = 2ab + c 2 a 2 + b 2 = c 2

3.22 – A SHORTER WAY Build a square off of each side of the following triangles (the first one is done for you.) Then find the area of each square. What is the relationship between the areas of the squares?

= 25

=169

=289

hagoreanv1.2.swf

PYTHAGOREAN THEOREM: (Peh-Tha-Gore-Ian) leg hypotenuse a b c a 2 + b 2 = c 2 If a triangle is a right triangle, then (leg) 2 + (leg) 2 = (hypotenuse) 2

3.23 – PYTHAGOREAN THEOREM For each triangle below, find the value of the variable. Write answers in simplified square root form.

y = 121 y 2 = 72 y = 11 2

= x 2 41 = x = x 2

x = 196 x 2 = 132 x = 14 2

x = 49 x 2 = 32