Answers:
- a = 18
- b = 36
- c = 18
- d = 18*sqrt(3)
sqrt is shorthand for "square root"
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Step-by-step explanation:
The smaller triangle on the left is a 45-45-90 triangle (aka isosceles right triangle). One special property of such triangles is that the legs are the same length, meaning a = c. Another special feature is that the hypotenuse is exactly sqrt(2) times that of the leg.
In other words, if x is the leg length, then x*sqrt(2) is the length of the hypotenuse. That means mean a = 18 in order for the hypotenuse to be 18*sqrt(2). This then leads to c = 18 as well.
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The smaller triangle on the right side has a = 18 as the shorter leg, opposite the smallest angle (30 degrees). The hypotenuse b is twice that of the short leg, and this works because we're dealing with a 30-60-90 triangle. Therefore b = 2a = 2*18 = 36.
The longer leg d is equal to sqrt(3) times the short leg, ie,
d = a*sqrt(3)
d = 18*sqrt(3)
You could use the pythagorean theorem to solve for d like so:
a^2 + d^2 = b^2
18^2 + d^2 = 36^2
d^2 = 972
d = sqrt(972)
d = 18*sqrt(3)
I skipped a bunch of steps, but you get the idea.