Answer
$A = 138.72$
Work Step by Step
Two similar triangles have corresponding sides that are proportional.
We know that the formula for the area of a rectangle is given as:
$A = bh$, where $A$ is the area, $b$ is the base, and $h$ is the height.
In $\triangle ABC$, $b = 4$, and $h = 3$. In $\triangle HIJ$, $b = x$ and $h = y$. We need to find the values for $x$ and $y$ to be able to find the area of this rectangle.
Let's take a look at one set of corresponding sides that have given values. We can take a look at the hypotenuses of both triangles. Let's set up a proportion to relate two sets of corresponding sides in the two triangles to solve for one of the variables:
$\frac{AC}{HJ} = \frac{AB}{HI}$
Let's plug in what we know:
$\frac{5}{17} = \frac{4}{x}$
Use the cross products property to eliminate fractions:
$5x = 68$
Divide both sides by $5$ to solve for $x$:
$x = 13.6$
Let's set up the proportion to solve for $y$:
$\frac{AC}{HJ} = \frac{BC}{IJ}$
Let's plug in what we know:
$\frac{5}{17} = \frac{3}{y}$
Use the cross products property to eliminate fractions:
$5y = 51$
Divide both sides by $5$ to solve for $x$:
$y = 10.2$
Now that we have the values for $x$ and $y$, we can plug them into the formula to find the area of the rectangle:
$A = (13.6)(10.2)$
Multiply to solve:
$A = 138.72$
