Answer:
30 square units
Step-by-step explanation:
The area of the given right triangle can be found a couple of ways:
Pythagorean theorem
The missing side length can be found using the Pythagorean theorem relation between the sides:
c² = a² +b²
13² = a² +5²
a² = 169 -25 = 144
a = √144 = 12
Then the area is ...
A = 1/2ab = 1/2(12)(5) = 30 . . . square units
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Trigonometry
The angle at lower right can be found from the trig relation ...
Cos = Adjacent/Hypotenuse
cos(A) = 5/13
A = arccos(5/13) ≈ 67.38°
The area can be found from these side lengths and the angle:
A = 1/2bc·sin(A)
A = 1/2(5)(13)sin(67.38°) = 30 . . . square units
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Additional comment
The sine of the angle can be computed from the cosine using the identity ...
sin(x) = √(1 -cos²(x))
In this case, the sine of the angle would be found to be 12/13, so the area would be computed as ...
A = 1/2(5)(13)(12/13) = 1/2(12)(5) . . . . same as using the Pythagorean theorem
