kite ABCD with angle A=60 degrees and angle C=90 degrees, diagonal DB=12 find the area and perimeter
In my diagram, angle C is at the top = 90°
then triangle BCD is an isosceles right-angled triangle with angle CBD = 45°
Let the diagonals intersect each other at E.
It is easy to see that EB = 6
then tan 45° = CE/6
CE = 6, (which we could have found by the isosceles triangle property)
by the 45-45-90 triangle ratios, BC = 6√2
look at triangel ABD, it is similar to the 30-60-90 triangle whose ratios of sides is 1 : √3 : 2
BA/2= 6/1 = AE/√3
AE = 6√3
AB = 12
perimeter = 6√2+6√2+12+12 = 24+12√2
area = (1/2)(12)(6) + (1/2)(12)(6√3) = 36 + 36√3
check my arithmetic
To find the area and perimeter of the kite, we need to use the given information and apply the formulas for these measurements.
Before we start, let's label the kite as shown:
D
/ \
/ \
A B
\ /
\ /
C
Now let's calculate the perimeter of the kite. The perimeter is the distance around the shape and can be found by adding up the lengths of all the sides.
Since angle C is a right angle (90 degrees), we can conclude that sides AD and BC are equal in length.
Since the diagonal DB is given as 12, we can also conclude that sides AB and CD are equal in length.
Let's label each side accordingly:
D___________E
/ \
/ \
A_________________B
\ /
\ /
C___________F
We can see that the diagonal DB divides the kite into two congruent right-angled triangles, ADB and BDC.
Using the Pythagorean theorem, we can find the lengths of sides AD and BC:
In triangle ADB:
AD^2 + AB^2 = DB^2
AD^2 + AB^2 = 12^2
AD^2 + AB^2 = 144
In triangle BDC:
BC^2 + AB^2 = DB^2
BC^2 + AB^2 = 12^2
BC^2 + AB^2 = 144
Since AD = BC and AB = CD, we can simplify the equations:
AD^2 + CD^2 = 144
2AD^2 = 144
AD^2 = 72
AD = BC = √72 ≈ 8.49 (rounded to two decimal places)
The perimeter of the kite is equal to the sum of all four sides:
Perimeter = AD + AB + BC + CD = 8.49 + 12 + 8.49 + 12 = 41.98 (rounded to two decimal places)
Now, let's find the area of the kite.
The area of a kite can be calculated using the formula:
Area = (diagonal1 * diagonal2) / 2
In this case, we only have one diagonal given: DB = 12
We need to find the length of the other diagonal.
Since angles A and C are given, we know that they are supplementary angles (A + C = 180 degrees). Therefore, angle B equals 180 - 60 - 90 = 30 degrees.
In triangle BDC, using the sine rule, we can find the length of segment CD:
sin(B) / DC = sin(C) / DB
sin(30) / DC = sin(90) / 12
1/2 / DC = 1 / 12
DC = (12 * 2) / 1 = 24
Now we have the lengths of both diagonals:
DB = 12
DC = 24
The area of the kite is:
Area = (DB * DC) / 2 = (12 * 24) / 2 = 144
Therefore, the area of the kite is 144 square units and the perimeter is 41.98 units.