1. A kite of area 40 m^2 has one diagonal 2 m longer than the other. Find the lengths of the diagonals.
2. Find the length of a side of an equilateral triangle of area 10.2 m^2.
3. A rhombus has an area of 40 cm^2 and adjacent angles of 50 degrees and 130 degrees. find the length of the side of the rhombus.
4. Find the area of a regular pentagon of side 8 cm.
5. A circle of radius 5 cm is inscribed inside a square as shown. Find the area shaded.
(the radius is not exactly 5 cm in the picture)
1. A kite of area 40 m^2 has one diagonal 2 m longer than the other. Find the lengths of the diagonals.
Areakite = D1 * D2 / 2 let D2 = D1 + 2
40 = D1 * [ D1 + 2] / 2 multiply both sides by 2
80 = D1 * [D1 + 2]
80 = [D1]^2 + 2D1 rearrange
[D1]^2 + 2D1 − 80 = 0 factor
( D1 + 10) (D1 − 8 ) = 0 set the factors to 0 → D1 = −10m [reject ] or D1 = 8 m
And D2 = D1 + 2 = 8 + 2 = 10 m
2. Find the length of a side of an equilateral triangle of area 10.2 m^2.
Area = (1/2)* side^2 * [√3 / 2 ]
10.2 = [√3/4 ] * side^2 multiply both sides by 4/ √3
[10.2 * 4 ] / √3 = side^2
40.8 / √3 = side^2 take the square root of both sides
√ [ 40.8 / √3 ] = side ≈ 4.853 m
5. A circle of radius 5 cm is inscribed inside a square as shown. Find the area shaded.
(the radius is not exactly 5 cm in the picture)
Shaded area = area of square − area of circle = [10 cm]^2 − pi [5cm]^2 =
100 cm^2 − 25 pi cm^2 = [ 100 − 25 pi] cm^2 ≈ 21.46 cm^2
It comes from the following "formula" for the area of an equilateral triangle
Area = (1/2) * side^2 * sin(60°) and sin(60°) = [√3 / 2 ]
So we have
Area = (1/2) * side^2 * [√3 / 2 ]
4. Find the area of a regular pentagon of side 8 cm.
Area ≈ 1.720 * side^2 = 1.720 * [8 cm]^2 = [1.720 * 64 ] cm^2 ≈ 110.08 cm^2
3. A rhombus has an area of 40 cm^2 and adjacent angles of 50 degrees and 130 degrees. find the length of the side of the rhombus.
We can use this :
(1/2) Area = (1/2) side^2 sin(50°) multiply by 2 on both sides
Area = side^2 sin(50°)
40 = side^2 sin(50°) divide both sides by sin(50)
40 / sin (50°) = side^2 take the sq root of both sides
√ [40 / sin (50°) ] = side ≈ 7.226 cm