Here are a few recommended readings before getting started with this lesson.
Try your knowledge on these topics.
Find the perimeter and area of the following figures.
A satellite orbiting the Earth uses radar to communicate with two stations on the surface. The satellite is orbiting in such a way that it is always in line with the center of Earth and Station From the perspective of Station the satellite is on the horizon. From the perspective of station the satellite is always directly overhead.
The measure of the angle between the lines from the satellite to the stations is To answer the following questions, assume that the Earth is a sphere with a diameter of kilometers. Write all the answers in kilometer, rounded to the nearest hundred.
Next, consider Station Because it is located on the Earth's surface, its distance from the center is equal to the radius of the Earth, which is kilometers.
Finally, the distance that a signal sent from Station to the satellite and then to Station is the sum of andThis number approximated to the nearest hundred is kilometers.
Approximate to nearest hundred
According to the Segment Addition Postulate, the length of — the distance that the signal will travel — can be found by adding andRounded to the nearest hundred the distance that the signal will travel is kilometers.
The heights of each pile will be calculated one at a time. Their difference can then be calculated.
The diameter of the pipes is twice their radius. Since the pipes are not staggered, they are directly on stacked on top of each other without a gap. Therefore, the height of the pile is the sum of the diameters of two vertically stacked pipes.
The height of the pile formed by the non-staggered pipes is centimeters.
To find the height of this pile, the triangle formed by the centers of two pipes next to each other and the pipe on top of them will be considered. Note that the length of each side of this triangle is equal to the sum of two radii. Therefore, the triangle is an equilateral triangle with a side length of centimeters.
Consider now the altitude of the above triangle. Note that the altitude of an equilateral triangle bisects the base. Recall also that the altitude of a triangle is perpendicular to the base. Therefore, a right triangle with hypotenuse centimeters and with side length of centimeters is obtained.
This information can be added to the diagram of the staggered pipes.
The height of the pile can be calculated by using the Segment Addition Postulate.
Calculate and compare the area of the three shapes.
The area of the three shapes will be calculated one at a time. Then, they will be compared.
Now, to find its area, the side length can be squared.
The three sides of an equilateral triangle have the same length. Therefore, to find the side length, the perimeter of the triangle, which is equal to the length of the fence, must be divided by To find the area of the triangle, its height must be found first. To do so, the altitude of the triangle will be drawn. Recall that the altitude of an equilateral triangle bisects and is perpendicular to the base.
Now that the areas of the three figures are known, they can be compared. To do so, the area of the triangle and the area of the circle will be approximated to two decimal places.
|Area of the Square||Area of the Equilateral Triangle||Area of the Circle|
It can be seen above that the circle is the figure with the greatest area. Therefore, Ali should construct Rover's playground in the shape of a circle. Run and feel the wind Rover!
The altitude of an equilateral triangle divides it into two right triangles. Use the Pythagorean Theorem to find the height of this triangle and then calculate its area. Finally, use the formula provided by the teacher to find the radius of Magdalena's circles and the radius of the incircle drawn by Dylan.
The area of the circles that Magdalena and Dylan drew will be calculated one at a time. Then, the results will be compared.
The circles will be ignored for a moment, and the altitude of the triangle will be drawn. The altitude of a triangle is perpendicular to the base. Also, because the triangle is equilateral, the altitude bisects the base. As a result, the length of one leg and the hypotenuse of the obtained right triangle are and centimeters, respectively.
Since the three circles are congruent, they have the same area. Therefore, to calculate the sum of the areas, it is enough to multiply the area of one of the circle's by
Calculating the sum of the areas of Dylan's circles takes less steps than calculating the sum of the areas of Magdalena's circles.
The sum of the areas of the circles that Magdalena and Dylan drew are known. For simplicity in the comparison, they will be approximated to one decimal place.
|Area of Magdalena's Circles||Area of Dylan's Circles|
It can be concluded that the circles drawn by Dylan have a greater area than the circles drawn by Magdalena.
In this lesson, different geometric methods were used to solve design problems. Here, the challenge presented at the beginning of the lesson will be examined in detail.Paulina bought two sprinklers to water her meter by meter rectangular garden. Each sprinkler waters a circular region, and the radius of each circle has the same measurement. Paulina can adjust the radius, which will affect both sprinklers equally. Recall that the sprinklers should not water any same region of the garden.
The area of the garden watered in each of the options will be calculated one at a time. Then, their difference will be found.