In order to find the area of the shaded region, we need to first find the areas of the triangle and the rectangle.

First, recall how to find the area of a triangle.

In the figure, we are given the triangle’s hypotenuse, which is also the width of the rectangle. We can then use this information and the Pythagorean theorem to find the length of the sides of the triangle.

Now, because this is a right isosceles triangle,

We can then make the following substitution:

Therefore:

Substitute in the value of the hypotenuse to find the length of the base:

Now, substitute this number in to find the area of the triangle.

Next, recall how to find the area of the rectangle:

Substitute in the given information from the question to find the area.

Now that we have the areas of both the rectangle and the triangle, we can find the area of the shaded region.

Solve.

In order to find the area of the shaded region, we need to first find the areas of the triangle and the rectangle.

First, recall how to find the area of a triangle.

In the figure, we are given the triangle’s hypotenuse, which is also the width of the rectangle. We can then use this information and the Pythagorean theorem to find the length of the sides of the triangle.

Now, because this is a right isosceles triangle,

We can then make the following substitution:

Therefore:

Substitute in the value of the hypotenuse to find the length of the base:

Now, substitute this number in to find the area of the triangle.

Next, recall how to find the area of the rectangle:

Substitute in the given information from the question to find the area.

Now that we have the areas of both the rectangle and the triangle, we can find the area of the shaded region.

Solve.

In order to find the area of the shaded region, we need to first find the areas of the triangle and the rectangle.

First, recall how to find the area of a triangle.

In the figure, we are given the triangle’s hypotenuse, which is also the width of the rectangle. We can then use this information and the Pythagorean theorem to find the length of the sides of the triangle.

Now, because this is a right isosceles triangle,

We can then make the following substitution:

Therefore:

Substitute in the value of the hypotenuse to find the length of the base:

Now, substitute this number in to find the area of the triangle.

Next, recall how to find the area of the rectangle:

Substitute in the given information from the question to find the area.

Now that we have the areas of both the rectangle and the triangle, we can find the area of the shaded region.

Solve.

First we need to know that a right isosceles triangle has two legs with the same length.

In this case we do not need to worry about the hypotenuse.

Since one leg measures 16 cm, the other leg is the same length and so our answer is 16cm.

Since  and  is a right angle,   is also a right angle.

  is the hypotenuse of the first triangle; since one of its legs  is half the length of that hypotenuse,  is 30-60-90 with  the shorter leg and  the longer. 

Because the two are similar triangles,  is the hypotenuse of the second triangle, and  is its longer leg.

Therefore, .

If all three angles of a triangle are congruent but the sides are not, then one of the triangles is a scaled up version of the other. When this happens the proportions between the sides still remains unchanged which is the criteria for similarity.

Though we must do a little work, we can show these triangles are similar. First, right triangles are not necessarily always similar. They must meet the necessary criteria like any other triangles; furthermore, there is no Hypotenuse-Leg Theorem for similarity, only for congruence; therefore, we can eliminate two answer choices.

However, we can use the Pythagorean Theorem with the smaller triangle to find the missing leg. Doing so gives us a length of 48. Comparing the ratio of the shorter legs in each trangle to the ratio of the longer legs we get

In both cases, the leg of the larger triangle is twice as long as the corresponding leg in the smaller triangle. Given that the angle between the two legs is a right angle in each triangle, these angles are congruent. We now have enough evidence to conclude similarity by Side-Angle-Side.

The Similar Figures Theorem holds that similar figures have both equal corresponding angles and proportional corresponding lengths. Either condition alone is not sufficient.  If two figures have both equal corresponding angles and equal corresponding lengths then they are congruent, not similar.

The Similar Figures Theorem holds that similar figures have both equal corresponding angles and proportional corresponding lengths. In other words, we need to know both the measures of the corresponding angles and the lengths of the corresponding sides. In this case, we know only the measures of  and . We don't know the measures of any of the other angles or the lengths of any of the sides, so we cannot answer the question -- they might be similar, or they might not be.

It's not enough to know that both figures are right triangles, nor can we assume that angles are the same measurement because they appear to be.

Similar triangles do not have to be the same size.

Since  and  are similar triangles, we know that they have proportional corresponding lengths.  We must determine which sides correspond.  Here, we know  corresponds to  because both line segments lie opposite  angles and between  and  angles. Likewise, we know corresponds to  because both line segments lie opposite  angles and between  and  angles.  We can use this information to set up a proportion and solve for the length of .

Substitute the known values.

Cross-multiply and simplify.

 and  result from setting up an incorrect proportion.  results from incorrectly multiplying  and .