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"The Median Theorem of Triangles" PPT Courseware 2
A and B are separated by a pond. Now we want to measure the distance between A and B, but we cannot measure it directly. What should we do? In this class, we will explore a method of measurement that seems impossible but can be accomplished.
As shown in the picture, choose a point C outside A and B to connect AC and BC, and find the midpoints D and E of AC and BC respectively. If the length of DE can be measured, can the distance of AB be known? .
Supplement: (1) Corollary to the theorem of bisectors of parallel lines
A straight line passing through the midpoint of one side of a triangle and parallel to the other side will bisect the third side.
Geometry language:
In △ ABC
∵ AD=DB，DE//BC
∴AE=EC
We call DE the median line of △ABC
Definition: The line segment connecting the midpoints of both sides of a triangle is called the median of the triangle.
The difference between the median line and the midline of a triangle:
The median of a triangle is the line segment connecting the midpoints of both sides of the triangle
The midline of a triangle is the line segment connecting a vertex to the midpoint of its opposite side
Understand the two meanings of the definition of the median line of a triangle:
① ∵D and E are the midpoints of AB and AC respectively.
∴DE is the median line of △ABC
② ∵ DE is the median line of △ABC
∴ D and E are the midpoints of AB and AC respectively.
A triangle has three median lines.
The median of a triangle is parallel to the third side and equal to half of the third side
Known: In △ABC, DE is the median line of △ABC
Prove: DE ∥ BC, and DE=1/2BC
Proof method 1.
Draw DE’∥BC through D, and intersect AC at point E’
∵D is the midpoint of side AB
∴E’ is the midpoint of AC (a straight line passing through the midpoint of one side of the triangle and parallel to the other side must bisect the third side)
So DE’ coincides with DE, so DE∥BC
Similarly, pass D to make DF∥AC, and cross BC to F
∴BF=FC= 1/2BC (A straight line passing through the midpoint of one side of the triangle and parallel to the other side must bisect the third side)
∴The quadrilateral DECF is a parallelogram
∴DE=FC ∴DE=1/2BC
Proof method 2.: As shown in the figure, extend DE to F, make EF=DE, and connect CF.
∵DE=EF, ∠AED=∠CEF, AE=EC
∴△ADE ≌ △CFE
∴AD=FC, ∠A=∠ECF
∴AB∥FC
And AD=DB ∴BD∥= CF
Therefore, quadrilateral BCFD is a parallelogram
∴DE ∥ BC and DE=1/2BC
1. Linked BD certificate: EH∥= FG
2. Connect AC and BD, certificate: EF∥HG, EH∥FG
3. Connect AC and BD, proof: EF=HG, EH=FG
⑴ Add the condition AC=BD to the quadrilateral ABCD, and the quadrilateral EFGH is _______. Why?
⑵ Add condition AC⊥BD to quadrilateral ABCD, what is quadrilateral EFGH _____? Why?
⑶If the quadrilateral EFGH is a square, what conditions should AC and BD satisfy?
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"The Median Theorem of Triangles" PPT courseware:
"The Median Line Theorem of a Triangle" PPT courseware to gain new knowledge. The line segment connecting the midpoints of both sides of a triangle is called the median line of a triangle. A triangle has three median lines. Comparison of different concepts between the median line of a triangle and the median line of a triangle (1) The similarities are all Relevant to the midpoint of the side..