Write a ratio in simplified form of the area of the shaded figure to the area of the figure which encloses it.
I have a square with height 5w and width 5w. Enclosing it are a right triangle on each side, which have a width of 6 and have the square's height (5w) also applying to these triangles.
no diagrams, and I can't see how two triangles the same height as the rectangle can enclose it.
To find the ratio of the area of the shaded figure to the area of the figure enclosing it, we need to calculate both areas.
Let's start by determining the area of the figure enclosing the shaded region. From the given information, we have a square with a height of 5w and a width of 5w. The area of a square is found by multiplying its length by its width. So, the area of the square is:
Area of the square = (5w) * (5w) = 25w^2.
Next, we will calculate the area of the shaded figure. According to the given description, the shaded figure is enclosed by a right triangle on each side. These triangles have a width of 6 and a height of 5w, which matches the height of the square. The formula to find the area of a right triangle is half the product of its base and height. Thus, the area of one triangle is:
Area of one triangle = (1/2) * (6) * (5w) = 15w.
Since there are four triangles enclosing the shaded figure, the total area of the shaded figure is:
Area of shaded figure = 4 * Area of one triangle = 4 * 15w = 60w.
To find the ratio of the area of the shaded figure to the area of the figure enclosing it, we divide the area of the shaded figure by the area of the figure enclosing it:
Ratio = Area of shaded figure / Area of enclosing figure = 60w / 25w^2.
Now, let's simplify the ratio. We can cancel out a common factor of "w" from the numerator and denominator:
Ratio = 60w / 25w^2 = (60 / 25w) * (w / w) = 12 / 5w.
Therefore, the ratio of the area of the shaded figure to the area of the figure enclosing it is 12/5w in simplified form.