# Find the area of the shaded region as a ratio to the area of the square

• chwala
In summary, the conversation discusses finding the area of a shaded region as a ratio to the area of a square using various methods such as geometry, trigonometry, and the shoelace formula. The final ratio is determined to be 1:8, with the area of the shaded region being 1/8 of the area of the square.
chwala
Gold Member

## Homework Statement

find the area of the shaded region as a ratio to the area of the square (kindly see attached diagram)

## The Attempt at a Solution

##A= \frac 1 2####b×h##
##A= \frac 1 2####×2x × 3x##

#### Attachments

• squareproblem.pdf
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chwala said:

## Homework Statement

find the area of the shaded region as a ratio to the area of the square (kindly see attached diagram)

## The Attempt at a Solution

##A= \frac 1 2####b×h##
##A= \frac 1 2####×2x × 3x##
Start by figuring out as many lengths as you can. The small upper right triangle is similar to the larger lower triangle, so the corresponding sides of these two triangles.

After you find all three sides of the small upper right triangle, you can calculate the angle that the two intersecting lines make, and from that, you can calculate the area of the shaded region. This problem is mostly geometry and a bit of trig.

Label the vertices of the square as A, B, C, D starting at top left and proceeding clockwise.
Label the other point on the top edge as E
Label the point of intersection of two lines inside the square as X.

Then the are of the shaded part is ##\Delta ABD - \Delta AED - \Delta EXB##.

The first two of those are easily calculated.
The base of the third triangle is known, so we only need to work out the perpendicular distance h from X to the top edge.

How do you think you might go about calculating that?

then ##(hX)^2= (EX)^2-(hE)^2## is this correct going with post 3.

chwala said:
then ##(hX)^2= (EX)^2-(hE)^2## is this correct going with post 3.
No. ##h## is a length, not a point. I think you interpreted it as meaning the point at the foot of the perpendicular from X to EB. But it is not that point, it is the height of the perpendicular. If we denote that foot by H (capital for a point, as opposed to lower case for a distance) and replace h by H in the equation you wrote, it is correct, but it doesn't lead clearly to a solution. I suggest you take h as the height mentioned, then work out the (perpendicular) distance from X to the line DC in terms of h. Then use the fact that triangle EXB is similar to CXD to write an equation in h, which you can then solve.

sysprog
using similarity
## \frac {h_{1}} {x}## = ##\frac {h_{2}} {3x}##
...Ratio of shaded : square is ## 1:8##

chwala said:
using similarity
## \frac {h_{1}} {x}## = ##\frac {h_{2}} {3x}##
...Ratio of shaded : square is ## 1:8##
Correct!

there's also an alternative method considering the square we have the equations ##y=x## and ##y=-3x+9## the point of intersection is ##(9/4,9/4)## now we have 4 points that is
##(0,0), (9/4,9/4), (2,3) and (0,0)##, using shoe lace formula, the area of shaded is ##\frac 9 8##units square and area of square is ##9## square units therefore,
ratios will be ##\frac 9 8##:##9## = ##\frac 1 8##:1 = ## 1:8##

sysprog

## 1. How do you find the area of the shaded region as a ratio to the area of the square?

In order to find the area of the shaded region as a ratio to the area of the square, you need to first calculate the area of the shaded region and the area of the square separately. Then, divide the area of the shaded region by the area of the square to get the ratio.

## 2. What is the formula for finding the area of a square?

The formula for finding the area of a square is A = s^2, where s is the length of one side of the square. This means that you need to multiply the length of one side by itself to get the total area of the square.

## 3. How do you calculate the area of a shaded region?

To calculate the area of a shaded region, you need to first identify the shape of the shaded region (e.g. triangle, circle, etc.) and then use the appropriate formula to find its area. Once you have the area of the shaded region, you can then follow the steps in question 1 to find the ratio to the area of the square.

## 4. Can the ratio of the shaded region to the area of the square be greater than 1?

Yes, the ratio of the shaded region to the area of the square can be greater than 1. This means that the shaded region is larger than the square and takes up more space.

## 5. How is finding the area of the shaded region as a ratio to the area of the square useful?

Calculating the area of the shaded region as a ratio to the area of the square can be useful in many real-life scenarios, such as calculating the percentage of a field that is being used for farming or the percentage of a pizza that has been eaten. It allows for easy comparison and understanding of the relative sizes of different areas.

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