You are planning to use a ceramic tile design in your new bathroom. The tiles are blue-and-white equilateral triangles. You decide to arrange the blue tiles in a hexagonal shape as shown. if the side of each tile measures 8 cm, what will the exact area of each hexagonal shape be? (1 point) Responses 963–√ cm2 963–√ cm2 1923–√ cm2 1923–√ cm2 1657 cm2 1657 cm squared 73.53–√ cm2
To find the ratio of the perimeters, we can simply find the ratio of the corresponding sides since when two triangles are similar, their sides are in proportion to each other.
Given that the widths are 12m and 8m, the ratio of the corresponding sides is 12m:8m which simplifies to 3:2.
Therefore, the ratio of the perimeters is 3:2.
To find the ratio of the areas of the similar triangles, we know that the ratio of the areas of similar figures is the square of the ratio of their corresponding sides.
Since the ratio of the corresponding sides is 3:2, the ratio of the areas will be (3/2)^2 = 9/4 = 36:16 = 18:8 which simplifies to 9:4.
Therefore, the ratio of the areas of the similar triangles is 9:4.
The area of a triangle is given by the formula:
Area = (1/2) * base * height
Given that the original area is 108, we can set up the equation:
108 = (1/2) * base * height
Next, if the height is reduced to one-fourth its original length and the base is quadrupled, we can represent the new base as 4 times the original base (4 * base) and the new height as one-fourth of the original height (1/4 * height).
Therefore, the new area can be calculated as:
New Area = (1/2) * (4 * base) * (1/4 * height)
New Area = (1/2) * (4 * base) * (1/4 * height)
New Area = (1/2) * base * height
As you can see, the new area is equal to the original area.
Therefore, the new area is 108.
So, the response "The area would not change" is correct.
If the tablecloth is 5 times wider and 5 times longer than the napkin, then the area of the tablecloth will be 5 times 5 = 25 times the area of the napkin, since area is length times width for rectangular shapes.
Since the cost is proportional to the area of the napkin, the cost for the tablecloth would be 25 times the cost of the napkin.
If the napkin costs $2.75, then the cost for the tablecloth would be:
25 * $2.75 = $68.75
Therefore, you would expect to pay $68.75 for the tablecloth.
Since the trapezoids are similar, the ratio of the areas of similar figures is the square of the ratio of their corresponding sides.
Let x be the height of the smaller trapezoid.
Since the base of the smaller trapezoid is 24 m, the height base ratio for the smaller trapezoid is 24: x.
The base of the larger trapezoid is 59 m, and the corresponding height (since the trapezoids are similar) will be 59: 5x.
The ratio of the bases is 59:24, and the ratio of the lengths of the trapezoids will be 59:24.
So, the ratio of the areas will be (59:24) ^ 2 = (59/24) ^ 2 = 3481 / 576.
Let the area of the larger trapezoid be A.
Since the area of the smaller trapezoid is 564 m^2, we have:
564 / A = 576 / 3481
A = (564 * 3481) / 576
A = 342708 / 576
A = 595.125
Rounded to the nearest whole number, the area of the larger trapezoid is approximately 595 m^2.
To find the area of the hexagonal shape, we need to find the area of each equilateral triangle and then multiply it by 6 (since there are 6 equilateral triangles in a hexagon).
The formula for the area of an equilateral triangle is:
Area = (√3/4) * s^2, where s is the side length of the triangle.
In this case, the side length of each equilateral triangle is 8 cm.
Plugging in the values:
Area = (√3/4) * 8^2
Area = (√3/4) * 64
Area = (√3 * 64)/4
Area = (64√3)/4
Area = 16√3 cm^2
Now, we need to multiply this by 6 to get the total area of the hexagon:
Total area = 16√3 * 6
Total area = 96√3
Therefore, the exact area of each hexagonal shape will be 96√3 cm^2.
The widths of two similar triangles are 12 m and 8 m. What is the ratio of the perimeters? Of the areas?(1 point) Responses 5 : 4 and 36 : 25 5 : 4 and 36 : 25 3 : 2 and 18 : 6 3 : 2 and 18 : 6 5 : 4 and 25 : 16 5 : 4 and 25 : 16 3 : 2 and 9 : 4
Find the area of the regular octogon with the given apothem of 10 yd. Round to the nearest tenth. (1 point) area =
To find the area of a regular octagon, you can use the formula:
Area = (1/2) * apothem * perimeter
Given that the apothem is 10 yards and the octagon is a regular octagon, you can find the perimeter of the octagon by recognizing that a regular octagon has 8 equal sides. It can be divided into 8 isosceles triangles, each having a base equal to one side of the octagon, and height equal to the apothem of the octagon.
Using trigonometry (in this case, with half of one of the isosceles triangles formed by the apothem and half of one side of the octagon), you can find the length of one side of the octagon:
tan(22.5 degrees) = (0.5 * s) / 10 yards
0.4142 = (0.5 * s) / 10
0.4142 * 10 = 0.5 * s
4.142 = 0.5 * s
8.284 = s
Now that you have the side length of the octagon (s = 8.284 yards), you can find the perimeter:
Perimeter = 8 * s
Perimeter = 8 * 8.284
Perimeter = 66.27 yards
Now, plug the values back into the formula to find the area:
Area = (1/2) * 10 * 66.27
Area = 5 * 66.27
Area = 331.35 square yards
Therefore, the area of the regular octagon with an apothem of 10 yards is approximately 331.35 square yards.
The area of a triangle is 108. If the height is reduced to one-fourth its original length and the base is quadrupled, what is the new area(1 point) Responses
4 4
27 27
The area would not change The area would not change
1/4