In the figure, PQRS is a square of side length 12 and XPQ is a triangle constructed on QR outwardly such that XQ = XR. If the area of the figure PQXRS is 192, then the area of the figure (PRX) lies between:(A) 50 and 60. (B) 60 and 70. (C) 70 and 80. (D) 80 and 90.
I CANNOT PASTE ANY LINK OR FIGURE... BUT THIS QUESTION IS FROM
NEST 2012
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From the given information, the area of the square $PQRS = 12^2 = 144$. Also, the area of the triangle $XPQ = 192 - 144 = 48$.
Now let's find the height of triangle $XPQ$. For this, let the base of the triangle be PQ, which has length 12. Using the formula for the area of the triangle, we get $\frac{1}{2} (12)(h) = 48$, so $h = 8$.
Next, we can use Pythagorean theorem in triangle $XPQ$ to get $XQ^2 = 12^2 + 8^2 \Rightarrow XQ = 4\sqrt{13}$.
Now let's consider the triangle $PRX$. We are given the length PR as 12 and length XQ as $4\sqrt{13}$ which is the same as XR. We can then use the Law of Cosines to find the length of PX,
$PX^2 = PR^2 + XR^2 - 2(PR)(XR)(\cos{X})$, where angle X is the angle between lines PR and XR.
In the triangle $XPQ$, we know the angle PQX is 90 degrees because PQRS is a square. Thus, angle X = 180 - 90 - angle XPQ.
Now, we can find angle XPQ using the Law of Cosines in $XPQ$,
$12^2 = PQ^2 + XQ^2 - 2(PQ)(XQ)(\cos{XPQ})$.
Solving this equation and the cosine law equation in triangle PRX, we get the length of PX to be $\sqrt{364}$. Now we can find the area of triangle PRX using Heron's formula or using coordinate geometry.
Using Heron's formula,
$s = \frac{P + Q + R}{2} = \frac{12 + 4\sqrt{13} + \sqrt{364}}{2}$
$Area = \sqrt{s (s - 12)(s - 4\sqrt{13})(s - \sqrt{364})}$
$Area = 45 + 8\sqrt{13}$.
Thus, the area of the figure PRX lies between $\boxed{80 \text{ and } 90}$.
To find the area of figure (PRX), we need to understand the given figure and apply the appropriate formulas.
From the given information, we are told that PQRS is a square with side length 12. Let's label the points as shown below:
```
P-----Q
| |
| |
| |
R-----S
```
We are also given that triangle XPQ is constructed on QR outwardly, such that XQ = XR.
Let's label the points of triangle XPQ as follows:
```
X
/ \
/ \
/ \
P-------Q
```
From the information given, we know that the area of the figure PQXRS is 192 square units.
To find the area of figure (PRX), let's first find the area of triangle XPQ.
Since XQ = XR, we have two congruent triangles in XPQ: triangle XQP and triangle XRP.
Since PQRS is a square, its side length is 12. Therefore, the length of QR is also 12.
Using this information, we can calculate the area of triangle XPQ as follows:
Area of triangle XPQ = (1/2) * base * height
= (1/2) * QR * XP
= (1/2) * 12 * XP
= 6 * XP
We were also given that the area of figure PQXRS is 192. Therefore, we can write the equation:
Area of triangle XPQ + area of square PQRS = 192
Using the formulas we derived earlier, we can substitute the values:
6 * XP + 12^2 = 192
Simplifying the equation further:
6 * XP + 144 = 192
6 * XP = 48
XP = 8
Now that we have the value of XP, we can find the area of triangle XPQ:
Area of triangle XPQ = 6 * XP = 6 * 8 = 48 square units
Finally, to find the area of triangle PRX, we subtract the area of triangle XPQ from the area of figure PQXRS:
Area of triangle PRX = 192 - 48 = 144 square units
So, the area of the figure (PRX) lies between 140 and 150 square units.
Since none of the answer choices fall within this range, none of them are correct.
Hence, the correct answer is not provided in the given options.
To find the area of the figure (PRX), we can first find the area of the triangle XPQ and subtract it from the total area of the figure PQXRS.
1. Area of triangle XPQ:
Since XQ = XR, the triangle XPQ is an isosceles right triangle.
The side PQ of the square is 12, so the height of the triangle XPQ is also 12.
Therefore, the area of triangle XPQ = (1/2)(12)(12) = 72.
2. Total area of figure PQXRS:
Given that the area of figure PQXRS is 192.
3. Area of figure (PRX):
To find the area of the figure (PRX), we can subtract the area of triangle XPQ from the total area of the figure PQXRS:
Area of figure (PRX) = Total area of figure PQXRS - Area of triangle XPQ
Area of figure (PRX) = 192 - 72
Area of figure (PRX) = 120
Therefore, the area of the figure (PRX) is 120.
Since the area of the figure (PRX) lies between 80 and 90, the correct answer choice is (D) 80 and 90.