直角梯形ABCD面积等分问题：求线段EF的长度

Alright, let's work through this right trapezoid problem step by step. First, let's clarify a key assumption here—since we're told the areas of EFCD and ABFE are equal, it's standard to assume EF is parallel to both AB and DC (without this, we can't solve the problem with the given info; this is a common setup for these trapezoid area split questions).

First, let's define our variables clearly:

• Let the height of the entire right trapezoid ABCD be ( h ) (this is the perpendicular distance between the two bases AB and DC).
• We know ( |AB| = 5 ), ( |DC| = 1 ), and ( S_1 = S_2 ).

Step 1: Calculate the total area of trapezoid ABCD #

The area of a trapezoid is the average of its two bases multiplied by its height. Let's compute that:

Total Area = (|AB| + |DC|) * h / 2 = (5 + 1) * h / 2 = 3h

Since ( S_1 ) and ( S_2 ) are equal, each must take up half the total area:

S_1 = S_2 = 3h / 2 = 1.5h

Step 2: Use the trapezoid area formula for the smaller quadrilaterals #

Since EF is parallel to the bases, both EFCD and ABFE are also trapezoids. Let's write their area equations:

• For EFCD: ( S_1 = (|DC| + x) * h_1 / 2 = 1.5h ) (where ( h_1 ) is the height of EFCD, the distance from DC to EF)
• For ABFE: ( S_2 = (x + |AB|) * h_2 / 2 = 1.5h ) (where ( h_2 ) is the height of ABFE, the distance from EF to AB)

We also know the sum of these smaller heights equals the total height of the original trapezoid: ( h_1 + h_2 = h ).

Step 3: Relate the smaller heights to the total height #

For a trapezoid with a line parallel to its bases, the ratio of each smaller trapezoid's height to the total height is proportional to the difference in their base lengths. That gives us:

h_1 / h = (x - |DC|) / (|AB| - |DC|) = (x - 1) / 4
h_2 / h = (|AB| - x) / (|AB| - |DC|) = (5 - x) / 4

Rearranging these gives ( h_1 = h(x - 1)/4 ) and ( h_2 = h(5 - x)/4 ).

Step 4: Solve for x #

Let's substitute ( h_1 ) into the area equation for EFCD. We can cancel out ( h ) from both sides (since height can't be zero):

(1 + x) * [h(x - 1)/4] / 2 = 1.5h

Simplify this equation:

(1 + x)(x - 1) / 8 = 1.5

Notice that ( (1 + x)(x - 1) = x^2 - 1 ), so we can rewrite this as:

x^2 - 1 = 1.5 * 8 = 12
x^2 = 13
x = √13 ≈ 3.606

We ignore the negative root because length can't be negative.

Quick shortcut formula #

If you're familiar with trapezoid properties, there's a handy formula for this scenario: when a line parallel to the bases splits a trapezoid into two equal areas, the length of that line ( m ) is:

m = √[(a² + b²)/2]

Where ( a ) and ( b ) are the lengths of the two bases. Plugging in ( a=1 ) and ( b=5 ):

m = √[(1² + 5²)/2] = √[(1 + 25)/2] = √13

This matches our earlier result, so we know we're correct.

内容的提问来源于stack exchange，提问作者user504613