Triangle $ABC$ has vertices $A(0,8)$, $B(2,0)$, $C(8,0)$. A vertical line intersects $AC$ at $R$ and $\overline{BC}$ at $S$, forming triangle $RSC$. If the area of $\triangle RSC$ is 12.5, determine the positive difference of the $x$ and $y$ coordinates of point $R$.
+2441 Constructing a diagram is a crucial step here; digesting all that information visually is extremely difficult without a visual aid. I have concocted one that I am proud of. I have tried to label everything in this diagram for your convenience. This problem most likely has many avenues toward achieving the correct answer, but I will present my approach for you:
In the diagram above, I have added point D, which is the intersection of point A and point C. The point lies on the origin. Let's focus on some characteristics regarding \(\triangle ACD\) .
Point A and point D have the same x-coordinate, so a vertical segment connects them. Point C and point D share the same y-coordinate, so they must be connected with a horizontal segment. \(\overline{AD}\perp\overline{DC}\) because horizontal and vertical lines are always perpendicular; thus, \(m\angle ADC=90^{\circ}\).
\(AD=DC=8\), so \(\triangle ACD\) is also an isosceles triangle.
A triangle that is both right and isosceles is also known as a 45-45-90 triangle. It is possible to observe these similarities with \(\triangle RSC\).
\(\overline{AD}\parallel\overline{RS}\) because they are both vertical lines. Whenever a parallel line cuts through a triangle, the resulting triangles are similar. In this case, \(\triangle ADC\sim\triangle RSC\) .
\(DC=8\) and \(DS=a\) , so \(CS=8-a\) . Because \(\triangle RSC\) is isosceles, \(RS=CS=8-a\) . The area of this triangle, according to the original problem, is 12.5. We know the lengths of the side lengths, so we can determine the value of a.
| \(12.5=\frac{1}{2}(8-a)(8-a)\) | Multiply by 2 on both sides to solve for a. We are using the area of a triangle to generate this equation. |
| \(25=(8-a)^2\) | Take the square root of both sides. |
| \(5=|8-a|\) | |
Yes, it is possible to solve for a here, and it is relatively simple from here on out. However, we can take a slight shortcut. The end goal is to determine the difference of x- and y-coordinates, but 8-a represents the height of the triangle. Since 8-a=5, we know that the y-coordinate is 5.
(3,5) is the coordinate of point R. \(|3-5|=2\) is the positive difference.
+128475 Thanks, X2 !!!....here's another approach.....
Let SC be the base of triangle RSC.....and let its length = 8 - x
The slope of the line segment AC = (0 - 8) / (8 - 0) = -8/8 = -1
And the equation of the line containing this segment is y = - x + 8 ⇒ y = 8 - x
So...let the height of triangle RSC = SR = 8 - x
And we have that the area of triangle RSC =
(1/2) (8 -x) (8 -x) = 12.5 multiply both sides by 2
(8 - x) ( 8 - x) = 25 simplify
x^2 - 16x + 64 = 25
x^2 - 16x + 39 = 0 factor
(x - 3) ( x - 13) = 0
Setting both factors to 0 and solving for x produces
x = 13 (reject) or
x = 3 (accept)
So.... since R lies on the line y = 8 - x, its coordinates are ( 3, 5)
And the positive difference of its x and y coordinates = 5 - 3 = 2
