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若一个四边形的一条对角线把四边形分成两个等腰三角形，我们把这条对角线叫这个四边形的和谐线，这个四边形叫做和谐四边形．如菱形就是和谐四边形．已知四边形ABCD中，...

Mar 9, 2024

若一个四边形的一条对角线把四边形分成两个等腰三角形，我们把这条对角线叫这个四边形的和谐线，这个四边形叫做和谐四边形．如菱形就是和谐四边形．已知四边形ABCD中，AB=AD=BC=2，∠BAD=90°，AC是四边形ABCD的和谐线，且AC≠CD，求四边形ABCD的面积

Solution by Steps

step 1

Given that AB=AD=BC=2 and ∠BAD=90°, triangle ABD is a right isosceles triangle

step 2

Using the Pythagorean theorem for triangle ABD, we have

AB^2 + AD^2 = BD^2

step 3

Substituting the given lengths,

2^2 + 2^2 = BD^2

, we find

BD^2 = 8

step 4

Taking the square root of both sides,

BD = \sqrt{8} = 2\sqrt{2}

step 5

Since AC is the diagonal that divides the quadrilateral into two isosceles triangles, and AC≠CD, triangle ABC is also isosceles with AB=BC=2

step 6

Using the Pythagorean theorem for triangle ABC, we have

AB^2 + BC^2 = AC^2

step 7

Substituting the given lengths,

2^2 + 2^2 = AC^2

, we find

AC^2 = 8

step 8

Taking the square root of both sides,

AC = \sqrt{8} = 2\sqrt{2}

step 9

The area of triangle ABD is

\frac{1}{2} \times AB \times AD = \frac{1}{2} \times 2 \times 2 = 2

step 10

The area of triangle ABC is the same as triangle ABD because they are congruent, so the area is also 2

step 11

The total area of quadrilateral ABCD is the sum of the areas of triangles ABD and ABC, which is

2 + 2 = 4

Answer

The area of quadrilateral ABCD is 4 square units.

Key Concept

Pythagorean theorem and properties of isosceles triangles

Explanation

The area of the quadrilateral was found by dividing it into two congruent right isosceles triangles and using the Pythagorean theorem to find the lengths of the diagonals, which are equal. The areas of the triangles were then calculated and summed to find the total area of the quadrilateral.