Given: ∆AKL, AK = 9
m∠K = 90o
m∠A = 60 °
Find: The perimeter of ∆AKL
The area of ∆AKL
for this one i got 81 for the perimeter
side a 6 and side b is 12
how about area?
P = 27 + 9(sqrt 3)
A = (81 sqrt 3) / 2
To find the perimeter of a triangle, you add up the lengths of all three sides.
In this case, you are given that AK = 9, and you need to find the lengths of KL and AL.
Since angle K is a right angle (90°), ∆AKL is a right triangle. From the given information, you know that angle A = 60°. This means that angle L is the complement of angle A, which is also 90° - 60° = 30°.
Since angle L is 30° and angle K is 90°, angle A must be 180° - (30° + 90°) = 60°.
Now, you can use the law of sines to find the lengths of KL and AL. The law of sines states that in any triangle, the ratio of the length of each side to the sine of its opposite angle is constant.
Using this formula:
sin(A) / a = sin(B) / b = sin(C) / c
You can set up the following proportions:
sin(30°) / KL = sin(60°) / 9
sin(30°) / AL = sin(90°) / 9
Simplifying these proportions:
(1/2) / KL = √3 / 9
(1/2) / AL = 1 / 9
Now, you can solve for KL and AL:
KL = (9 * √3) / (2 * 2)
= (9 * √3) / 4
AL = (9 * 1) / (2 * 2)
= 9 / 4
Finally, you can calculate the perimeter of ∆AKL:
Perimeter = AK + KL + AL
= 9 + (9 * √3) / 4 + 9 / 4
To calculate the area of ∆AKL, you can use the formula for the area of a right triangle:
Area = (base * height) / 2
Since AK is the base and KL is the height, you can plug in the values:
Area = (9 * (9 * √3) / 4) / 2
Simplifying and evaluating the expression will give you the final answers for both the perimeter and the area of ∆AKL.
in a 30-60-90 right triangle, the ratio of the sides is 1:√3:2
m∠L = 30°, so the short leg is 9. The sides are thus
9,9√3,18 and the perimeter is 27+9√3
Looks like you need to review the standard triangles...
for the area i got 36
(right?)