Question

what is the area figure

Answer 1

The area of a triangle is:
$A= \frac{1}{2}bh$

To solve for the base, you'll have to use trigonometric ratios. Since thee base is the opposite of the angle we are given, the trigonometric ratio that deals with opposite is sin(x), which is opposite/hypotenuse. So, let's set it up (x is the base):
$sin(45) = \frac{x}{24}$

Solve for x:
$24sin(45) = x$

Simplify (sin of 45 is a common ratio):
$x = \frac{24 \sqrt{2} }{2} = 12 \sqrt{2}$

Then, for the height, it's the side adjacent to angle given. We could use the Pythagorean theorem, but it's much easier and faster to use another trigonometric ratio. Unless you are told to use the Pythagorean Theorem, in which case tell me. So, the trigonometric ratio that deals with adjacent and hypotenuse is cos(x), which is adjacent over hypotenuse. Let's set it up (x is the height):
$cos(45) = \frac{x}{24}$

Simplify and solve for x:
$x = 24cos(45)$

Simplify (cos of 45 is also a common ratio):
$x = \frac{24 \sqrt{2} }{2} = 12 \sqrt{2}$

So, now we are ready to solve for the area. We know the base and the height! Let's plug everything in the formula:
$A = \frac{1}{2}(12 \sqrt{2})(12 \sqrt{2}) = \frac{1}{2}(144)(2) = 144$

So, the area of the triangle is 144 ft²

For these types of problems, you need to know all the trigonometric ratios. A good way to memorize it is using a mnemonic device like SohCahToa. The Soh stand for "Sine Opposite Hypotenuse", the Cah stands for "Cosine Adjacent Hypotenuse" and the Toa stands for "Tangent Opposite Adjacent". There are a lot more trigonometric ratios including inverse trigonometric ratios, reciprocal trigonometric ratios, hyperbolic trigonometric ratios (which show up during differential calculus), and the inverse and reciprocal of hyperbolic trigonometric ratios. Hope my answer helps! If anything doesn't make sense, please make sure you tell me!