Answer:
1. Length of BC = 35
2. SU = 3.5
3. AC = 50 inches
4.
Part 1: AB = 10 cm
Part 2: AD = 2 cm
5. Width of River AB = 145.45 ft
Step-by-step explanation:
1.
The length of BC is (x+4)+(2x+1) = 3x+5
Now, if we figure out x, we can plug that in and find length of BC.
Using similarity with the two triangles shown, we can set-up the ratio as:
[tex]\frac{8}{12}=\frac{x+4}{2x+1}[/tex]
Now, cross multiplying and solving for x:
[tex]\frac{8}{12}=\frac{x+4}{2x+1}\\8(2x+1)=12(x+4)\\16x+8=12x+48\\4x=40\\x=\frac{40}{4}=10[/tex]
Now plugging in x=10 into 3x+5, we have 3(10)+5 = 35
Length of BC = 35
2.
Using pythagorean theorem in triangle PQT, we can solve for QT.
[tex](\sqrt{2})^2+(\sqrt{2})^2=QT^2\\2+2=QT^2\\4=QT^2\\QT=\sqrt{4}=2[/tex]
QT = RS = 2
Now using pythagorean theorem on Triangle RSU, we can solve for SU. So:
[tex]RS^2+SU^2=RU^2\\2^2+SU^2=4^2\\SU^2=4^2-2^2\\SU^2=12\\SU=\sqrt{12}=3.5[/tex]
SU = 3.5
3.
If we draw a straight line as Segment AC, we have a right triangle with both legs measuring 30 and 40 inches, respectively. AC is the hypotenuse. Using pythagorean theorem, we can find out AC:
[tex]AB^2+BC^2=AC^2\\30^2+40^2=AC^2\\2500=AC^2\\AC=\sqrt{2500}=50[/tex]
Thus AC = 50 inches
4.
AB:
We can set-up a similarity ratio to solve for AB. We can write:
[tex]\frac{12}{20}=\frac{6}{AB}[/tex]
Now, cross multiplying, we can solve for AB:
[tex]\frac{12}{20}=\frac{6}{AB}\\12AB=6*20\\12AB=120\\AB=\frac{120}{12}=10[/tex]
Thus, AB = 10 cm
AD:
We know, AB = BF + FD + DA
We also know, FD = 6, AB = 10 and BF & DA are same. So we can write DA in place of BF and solve. Thus:
[tex]AB = BF + FD + DA\\10=AD+6+AD\\10-6=2AD\\4=2AD\\AD=2[/tex]
Thus, AD = 2 cm
5.
A single piece of information is missing from this problem. They have given DE = 32 ft.
Now, we see that triangle EDC is similar to triangle ABC, so their corresponding sides are proportional. Thus we can set-up a ratio as:
[tex]\frac{DC}{BC}=\frac{DE}{BA}[/tex]
Now we can put the information we know and solve for AB, the width of the river.
[tex]\frac{DC}{BC}=\frac{DE}{BA}\\\frac{22}{100}=\frac{32}{AB}\\22AB=32*100\\22AB=3200\\AB=\frac{3200}{22}=145.45[/tex]
Width of River AB = 145.45 ft
