菱形

菱形是平行四边形，其四个边的长度都相同。也称为等边四边形。该图代表菱形。

特性

• 菱形的所有面都是平等的。
• 菱形的相对两侧是平行的。
• 菱形的相对角度相等。

菱形对角线的性质

• 在菱形中，对角线会以直角彼此一分为二。
• 对角线将菱形的角一分为二。

证明

1.如果四边形是菱形，则对角线是彼此垂直的等分线。

首先，我们将证明对角线一分为二，然后我们将证明对角线彼此垂直。

(a)对角线一分为二

证明：

Since the rhombus is a parallelogram so the opposite sides are parallel and equal to each other.

So in the triangle AOD and COB we have 

AD = CB ( opposite sides)

∠ADO = ∠CBO (opposite angles)

∠DAO = ∠BCO (opposite angles)

Hence both the triangles are congruent.

Hence AO = CO and DO = BO.

Hence the diagonals bisect each other.

(b)对角线彼此垂直

证明：

Given that ABCD is a rhombus 

AB = AD                     (definition of rhombus)

AO = AO                    (Common side)

BO = OD                    (diagonals bisect each other as proved above)

△AOD ≅ △AOB   (Side-Side-Side postulate)      

∠AOD ≅ ∠AOB    (Corresponding angles in congruent triangles )

Also ∠AOD + ∠AOB = 180

Hence AC ⊥ DB.

Hence the diagonals are the perpendicular bisector of each other.

2.菱形的面积等于对角线乘积的一半。

证明：

Given that ABCD is a rhombus the diagonal AC and BD cut at point O.

Now we know that ∠AOD = ∠AOB = ∠COD = ∠BOC = 90 degree.

Hence we can see that the rhombus ABCD is equally into four parts.

So the area of rhombus ABCD = area of triangle AOD + area of triangle AOB + area of triangle BOC + area of triangle COD.

1/2 × AO × OD + 1/2 × AO × OB + 1/2 × BO × OC + 1/2 × OD × OC

= 1/2 × AO (OD + OB) + 1/2 × OC (BO + OD)

= 1/2 × (OD + OB) × (AO + OC)

Hence the area of the rhombus is equal to half of the product of diagonals.

例子

示例1：计算斜线长度为20厘米和10厘米的菱形的面积?

解决方案：

We are Provided that 

d1 = 20 cm

d2 = 10 cm

Area of a rhombus, A = (d1 × d2) / 2

= (20 × 10) / 2

= 100 cm²

Hence, the area of a rhombus is 100 cm².

例2：找出菱形的面积，每边等于25厘米，对角线之一的长度等于14厘米?

解决方案：

ABCD is a rhombus in which AB = BC = CD = DA = 25 cm

AC = 14 cm

Area of rhombus = 1/2 * d₁ * d₂

Therefore, BO = 7 cm

In ∆ AOB,

AB² = AO² + OB²

⇒ 25² = 7² + OB²

⇒ 625 = 49 + OB²

⇒ 576 = OB²

⇒ OB = 24

Therefore, BD = 2 x OB = 2 × 24 = 48 cm

Now, area of rhombus = 1/2 × d₁ × d₂ = 1/2 × 14 × 48 = 336 cm²

Hence the area of the rhombhus is is 336 cm².

示例3：求出面积为400平方米，周长为160平方米的菱形的高度?

解决方案：

Given, the perimeter of the rhombus = 160 m

So, side of rhombus = 160/4 = 40 m

We know that the area of any parallelogram = b × h

Therefore the height is:

⇒ 400 = 40 × h

⇒ h = 10 m

Therefore, the height of the rhombus is 10 m.