Areas of Parallelograms and Triangles Skill 33

Objective HSG-MG.1/7: Students are responsible for finding the areas of parallelograms and triangles.

Definitions A base of a parallelogram can be any one of its sides. The corresponding altitude is a segment perpendicular to the line containing the base, drawn from the side opposite the base. The height is the length of the altitude. The base of a triangle can be any of its sides. The corresponding altitude is the length of the altitude to the line containing the base.

Thm. 61: Area of a Rectangle The area of a rectangle is the product of a base and height. Thm. 62: Area of a Parallelogram The area of a parallelogram is the product of a base and the corresponding height. Thm. 63: Area of a Triangle The area of a triangle is the half the product of a base and the corresponding height.

Example 1; Finding Angle Measures of Trapezoids a) 𝐶𝐷𝐸𝐹 is an isosceles trapezoid and 𝑚∠𝐶=65ᵒ. What are 𝑚∠𝐷, 𝑚∠𝐸, and 𝑚∠𝐹? 𝒎∠𝑪+𝒎∠𝑫=𝟏𝟖𝟎 𝟔𝟓+𝒎∠𝑫=𝟏𝟖𝟎 E F C D 𝒎∠𝑫=𝟏𝟏𝟓ᵒ 𝒎∠𝑪=𝒎∠𝑭=𝟔𝟓ᵒ 65ᵒ 𝒎∠𝑫=𝒎∠𝑬=𝟏𝟏𝟓ᵒ

Example 1; Finding Area of a Parallelogram b) 4.6 cm 3.5 cm 2 cm 4.5 in. 4 in. 5 in. 𝑨=𝒃𝒉 𝑨=𝒃𝒉 𝑨=𝟓 𝟒 𝑨=𝟐 𝟑.𝟓 𝑨=𝟐𝟎 𝒊𝒏 𝟐 𝑨=𝟕 𝒄𝒎 𝟐 c) 10 m 9 m 12 m 𝑨=𝒃𝒉 𝑨=𝟏𝟐 𝟗 𝑨=𝟏𝟎𝟖 𝒎 𝟐

Example 2: Find the Missing Dimension a) Find D𝐸 to the nearest tenth. 9 in. 13 in. 9.4 in. A B C D E 𝑨=𝒃𝒉 𝑨=𝟏𝟑 𝟗 𝑨=𝟏𝟏𝟕 𝒊𝒏 𝟐 𝑨=𝒃𝒉 𝟏𝟏𝟕=𝟗.𝟒 𝑫𝑬 𝑫𝑬=𝟏𝟐.𝟒 𝒊𝒏.

Example 2: Find the Missing Dimension b) A parallelogram has sides 15 cm and 18 cm. The height corresponding to a 15-cm base is 9 cm. What is the height corresponding to an 18-cm base? Sketch the situation to help you. 𝑨=𝒃𝒉 15 cm 18 cm A B C D E 𝑨=𝟏𝟓 𝟗 9 cm 𝑨=𝟏𝟑𝟓 𝒄𝒎 𝟐 𝑨=𝒃𝒉 𝟏𝟑𝟓=𝟏𝟖 𝑫𝑬 𝑫𝑬=𝟕.𝟓 𝒄𝒎

Example 3; Finding Area of a Triangle a) Find the area of the triangle at the left. In square feet. 𝒉=𝟏𝟐𝒇𝒕∗ 𝟏𝟐𝒊𝒏 𝟏𝒇𝒕 +𝟐𝒊𝒏 𝒃=𝟏𝟒𝒇𝒕∗ 𝟏𝟐𝒊𝒏 𝟏𝒇𝒕 +𝟒𝒊𝒏 𝒉=𝟏𝟒𝟔 𝒊𝒏 𝒃=𝟏𝟔𝟎 𝒊𝒏 𝑨= 𝟏 𝟐 𝒃𝒉 = 𝟏 𝟐 𝟏𝟔𝟎 𝟏𝟒𝟔 12ft 2in. 13ft 4in. 𝑨=𝟏𝟏𝟔𝟖𝟎 𝒊𝒏 𝟐 𝟏𝟏𝟔𝟖𝟎 𝒊𝒏 𝟐 ∗ 𝟏𝒇𝒕 𝟏𝟐𝒊𝒏 ∗ 𝟏𝒇𝒕 𝟏𝟐𝒊𝒏 𝟖𝟏 𝟏 𝟗 𝒇𝒕 𝟐

Example 3; Finding Area of a Triangle b) Find the area of the triangle at the left. In square inches. 𝒃=𝟏𝟐 𝒊𝒏 𝒉=𝟓 𝒊𝒏 1 ft. 1 ft. 1 in. 5 in. 𝑨= 𝟏 𝟐 𝒃𝒉 = 𝟏 𝟐 𝟏𝟐 𝟓 𝑨=𝟔𝟎 𝒊𝒏 𝟐

Example 4; Finding Area of the Irregular Figure a) What is the area of the figure at the right in square inches? 𝑨= 𝟏 𝟐 𝒃𝒉 8 in. 6 in. 𝑨= 𝟏 𝟐 𝟔 𝟖 𝑨=𝟐𝟒 𝒊𝒏 𝟐 𝑨=𝒃𝒉 𝑨= 𝟔 𝟔 𝑨=𝟑𝟔 𝒊𝒏 𝟐 𝑨=𝒕𝒓𝒊𝒂𝒏𝒈𝒍𝒆+𝒔𝒒𝒖𝒂𝒓𝒆 𝑨= 𝟐𝟒 𝟑𝟔 𝑨=𝟔𝟎 𝒊𝒏 𝟐

Example 4; Finding Area of the Irregular Figure b) What is the area of the figure at the right in square inches? 7 in. 5 in. 𝑨= 𝟏 𝟐 𝒃𝒉 𝑨= 𝟏 𝟐 𝟓 𝟕 𝑨=𝟏𝟕.𝟓 𝒊𝒏 𝟐 𝑨=𝒃𝒉 𝑨= 𝟓 𝟓 𝑨=𝟐𝟓 𝒊𝒏 𝟐 𝑨=𝒕𝒓𝒊𝒂𝒏𝒈𝒍𝒆𝒔+𝒔𝒒𝒖𝒂𝒓𝒆 𝑨=𝟒 𝟏𝟕.𝟓 𝟐𝟓 𝑨=𝟗𝟓 𝒊𝒏 𝟐

#33: Areas of Parallelograms and Triangles Questions? Summarize Notes Homework Quiz