Given: ∆AKL, AK = 9
m∠K = 90o
m∠A = 60 °
Find: The perimeter of ∆AKL
The area of ∆AKL
1/ AK perpendicular to LK
2/AL is the third side
plz add answer and solution
this is a 30-60-90 triangle
m∠L = 30 °
... AK is the side opposite
AL (the hypotenuse) is twice AK
KL is AK times √3
the area is 1/2 the product of the two perpendicular sides
To find the perimeter of ∆AKL, we need to find the lengths of all three sides and add them together.
Given that AK = 9 and ∠A = 60 °, we can use trigonometry to find the length of AL, the third side of the triangle. In a right triangle, where one angle is 90 °, the sides are related by the Pythagorean theorem: a^2 + b^2 = c^2.
In this case, AK is the shorter side (a) and AL is the longer side (c). Applying the Pythagorean theorem, we have:
9^2 + AL^2 = c^2
Simplifying:
81 + AL^2 = c^2
Since we already know that ∠K = 90 °, we can use the sine function to find the length of AL:
sin(60 °) = AK / AL
Rearranging the equation:
AL = AK / sin(60 °)
AL = 9 / sin(60 °)
AL ≈ 10.39
Now we have the lengths of all three sides:
AK = 9
AL ≈ 10.39
LK (hypotenuse) = AK
The perimeter of ∆AKL is the sum of these three sides:
Perimeter = AK + AL + LK
Perimeter = 9 + 10.39 + 9
Perimeter ≈ 28.39
Therefore, the perimeter of ∆AKL is approximately 28.39.
To find the area of ∆AKL, we can use the formula for the area of a right triangle: Area = (base * height) / 2.
In this case, AK is the base and AL is the height. Therefore:
Area = (AK * AL) / 2
Area = (9 * 10.39) / 2
Area ≈ 46.95
Therefore, the area of ∆AKL is approximately 46.95.