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math chapter 12
Math ch. 12 area of rectangles
| Question | Answer |
|---|---|
| area | to be completely covered without gaps, L x W (L groups of square and W squares in each group- a two dimensional attribute) |
| What are the two principles that are used in determining the area of a shape? | Moving Principle and Additivity Principles |
| Moving Principle | If you move a shape rigidly without stretching it, then its area does not change |
| Additivity Principle | If you combine ( a finite number of) shapes without overlapping them, then the area of the resulting shape is the sum of the areas of the individual shapes. |
| What do the Moving principle and Additivity Principle allow you to do with shapes? | Allows us to take shapes apart and calculate areas of the pieces |
| How can you use the Additivity Principle to find the area of some shape? | Subdivide the shape into pieces whose areas are easy to determine then add the areas of these pieces. The resulting sum is the area of the original shape because the original shape is the combination of its' pieces. |
| How can you use the Moving and Additivity Principles to determine the area of a shape? (p519) | Subdivide the shape into pieces, then move and recombine those pieces, without overlapping, to make a new shape whose area is easy to determine. (like if there was a bump sticking out move it into the rectangle) |
| How else can the Additivity Principle be sued when there is an area you want to leave out- shaded area? (p520) | You can use the additivity principle to "take away" an area. For example when there is a little rectangle in the middle of a large rectangle that is shaded. Find the area of the small and the large rectangles and subtract the small one out. |
| What principles can be used to determine areas of triangles? | the Moving and Additivity Principles |
| What is the most primitive way to determine the area of a triangle? | count how many 1 unit by 1 unit squares it takes to cover the shape without gaps or overlaps. This method requires square to be cut apart and pieces to be moved and recombined with other pieces to make as many full squares as possible. |
| What is a more sophisticated way of determining the area of a triangle? (2 ways) | Relating the triangle to a rectangle either by moving a big chunk or by embedding the triangle in a rectangle. These methods lead to the triangle area formula |
| If we generalize the more sophisticated ways of determining the area of a triangle we arrive at an explanation for the formula of the area of a triangle which is? | one-half the base times the height |
| base | base of a triangle can be any one of its three sides, "b" |
| what does "b" really mean ? | length of the base |
| height | the line segment that is perpendicular to the base and connects the base or an extension fo the base to the vertex of the triangle that is not on the base , "h" (see fig12.20 p. 524 |
| Does it matter which side is chosen to be the base? | no. No matter which side is chosen the formula works. The height just needs to correspond |
| Formula for Area of a triangle with base b and height h | 1/2 (b x h) square units |
| If the base is 5 inches long and the height is 3 inches long what is the area of the triangle? | 1/2 (5 x3) inches squared = 1/2 (15) = 7.5 inches squared |
| Why is the area formula valid or how do you prove it? | two copies of the triangle can be subdivided and then recombined without overlapping to form a "b" by "h" rectangle. Because of moving and additivity principles about area |
| So how do you prove it using formulas that the area formula is valid | 2 x area of triangle = area of rectangle. By substituting b x h for the area of the rectangle 2 x area of triangle = b x h : therefore area of triangle = 1/2 (b x h) |
| If you have an unusual shaped triangle and you can't really turn it into a half b x h rectangle how would you prove the formula for the area of a triangle works? | To explain the area enclose the triangle in a rectangle. figure out the rectangles area. put the two shaded triangles together they form a rectangle of area. Take away the two triangle areas to get the original triangle area |
| What is the formula involved for proving area of a triangle by putting it inside a rectangle? (p526) | rectangle area (b+a) x h which is equal to b x h = a x h (distributive property), area of two triangles is a x h . Take away a x h (area of two triangles) from area of rectangle to get (bxh + axh) - axh = bxh so original triangle is 1/2 this area 1/2bh |
| base of a parallelogram | also b like rectangle |
| height of a parallelogram | "h" It is perpendicular to the base, it connects the base or an extension of the base to vertex of the base |
| Formula for the area of a parallelogram with base b units long and height h units long | b x h square units |
| Why is the formula for a parallelogram valid or how can you prove it? | subdivide the parallelogram and recombine it to form a b x h rectangle. according to the moving and additivity principle about area the area of the parallelogram and the new rectangle are equal. b x h square units |
| How do you show the formula for a parallelogram by running a straight line down to form triangles? p. 534 | enclose the parallelogram in a rectangle. the rectangle consists of the parallelogram and two copies of a right triangle. Use distributive property on the area of rectangle, put triangles together and take away from the rectangle |
| What are the formulas for subdividing and recombining a parallelogram to make a rectangle proving the are of parallelogram formula? | rectange (a+b) x h same as a x h=b xh. two triangles make area a x h. Take a x h away from big rectangle..(bxh = axh) - axh = b x h |
| Area of a circle | A=p r squared |
| Pi | 3.14159 |
| Circumference | The distance around the circle |
| Circumference divided by diameter equals ? | Pi. For any circle |
| Circumference formula | C = pi D. Or c = 2 pi r |
| Radius | 1/2 the diameter, the distance from the center of the circle to a point on the circumference of the circle |
| Pythagorean Theorem | A squared + b squared = c squared |
| Hypotenuse | The side opposite the right angle in a triangle |
| Proof | Thorough, precise, logical explanation for why a statement is true |
| What principles can you use to prove the Pythagorean theorem? | The moving and additive to principles |
| In general how do you prove the Pythagorean theorem is correct using the moving and additive to principles? | Form 4 triangles (making 2 rectangles) that forms a large square (sides b) and small square (sides a). Other large square is made up of 4 triangles on all sides with large square (sides c) tilted in center |
| How do you explain the Pythagorean theorem with the two large squares that have the 4 triangles in each large square? | Remove |
Created by:
murphy1717