a Recall that in similar triangles corresponding sides are proportional.
B
b Start with evaluating the area of the shaded regions.
C
c Notice that in similar triangles corresponding angles are congruent.
A
a DF=32 in.
Explanation: See solution.
B
b Dark fabric: 820 in^2
Light fabric: 180 in^2
C
c Yes, see solution.
Practice makes perfect
a We are given that a youth organization is designing a rectangular flag with two shaded triangles. Let's take a look at the given diagram.
Since the designers want the shaded triangles to be similar, the corresponding sides of these triangles need to be proportional. Having this in mind, we can write that the ratio of the shorter legs in these triangles is equal to the ratio of their longer legs.
ED/CB=DF/BA
Next, by substituting appropriate side lengths, we will solve for DF.
b In this part we are asked to find how much dark and light fabric designers need. To do this, we need to evaluate the areas of the shaded and non-shaded regions. Let's start with the area of the shaded triangles.
As we can see, the area of the shaded region will be the sum of the areas of △ ABC and △ EDF. Recall that the area of a right triangle is half of the product of its legs. With this information, we can evaluate the area of each shaded triangle.
A_(ABC)=1/2( 25)(40)=500
A_(EDF)=1/2(20)(32)=320
Next, we should add these areas to find how much dark fabric is needed by designers.
500+320=820
Designers need 820 square inches of dark fabric. Now we will find how much light fabric designers need. To do this, we can evaluate the area of the whole flag and then subtract the area of the shaded region, 820. Recall that the area of a rectangle is a product of its dimensions.
We found that the designers need 180 square inches of light fabric.
c In this part we are asked to determine whether the hypotenuses of these two triangles are parallel. First, recall that in similar triangles corresponding angles are congruent. Let's mark this fact on the given diagram.
Now we can see that the hypotenuses intersect transversal DC at the same angle. Therefore, by the Converse of the Corresponding Angles Theorem, the hypotenuses are parallel.
