People's Education Press Ninth Grade Mathematics Volume 2

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"Similar Triangles" Similar PPT Courseware

1. Similar graphics

Definition: Figures with the same shape are called similar figures.

2. Proportional line segment

Definition: Among the four line segments a, b, c, d, if the ratio of two of the line segments is equal to the ratio of the other two line segments, that is, ab=cd (or a:b=c:d), then the four line segments a, b, c, and d are called proportional line segments, or simply proportional line segments.

Note: (1) Line segments a, b, c, d are proportional and sequential, indicating ab=cd (or a∶b=c∶d);

3. Properties of proportional line segments

Properties: (1) Basic properties: If a∶b=c∶d or ab=cd, then ad=bc; in particular, if a∶b=b∶c or ab=bc, then b2=ac.

(2) Composite property: If ab=cd, then a±bb=c±dd.

4. Similar polygons

Definition: Two polygons whose corresponding angles are equal and whose corresponding sides are proportional are called similar polygons.

Note: Two polygons that only correspond to proportional sides are not necessarily similar, such as rhombuses; two polygons that only correspond to equal angles are not necessarily similar, such as rectangles.

Similarity ratio: The ratio of corresponding sides of similar polygons is called similarity ratio.

5. Similar triangles

Definition: Triangles whose corresponding angles are equal and whose corresponding sides are proportional are called similar triangles.

Judgment: (1) The straight line parallel to one side of the triangle intersects the other two sides (or extension lines), and the triangle formed is similar to the original triangle;

(2) If the ratios of the three corresponding sides of two triangles are equal, then the two triangles are similar;

(3) If the ratios of the two corresponding sides of two triangles are equal and the corresponding angles are equal, then the two triangles are similar;

(4) If two angles of one triangle are equal to two angles of another triangle, then the two triangles are similar;

(5) If the ratios of the hypotenuse and a right side of two right triangles are equal, then the two right triangles are similar.

Note: The two right triangles divided by the height of the hypotenuse are similar to the original triangle.

[2010·Zhuhai] As shown in Figure 38-1, in the parallelogram ABCD, AE⊥BC passes through the point A, the vertical foot is E, connecting DE, F is a point on the line segment DE, and ∠AFE=∠B.

(1) Verify: △ADF∽△DEC;

(2) If AB=4, AD=33, AE=3, find the length of AF.

[Analysis] (1) Prove that ∠AFD=∠C, ∠ADF=∠CED;

(2) From △ADF∽△DEC, we get ADDE=FACD, and AD, DE, and CD are known or can be found, so it is easy to find FA.

Solution: (1) Prove: ∵ Quadrilateral ABCD is a parallelogram,

∴AD∥BC, AB∥CD,

∴∠ADF=∠CED, ∠B+∠C=180°.

∵∠AFE+∠AFD=180°, ∠AFE=∠B,

∴∠AFD=∠C, ∴△ADF∽△DEC.

(2)∵ Quadrilateral ABCD is a parallelogram,

∴AD∥BC，CD=AB=4.

Also ∵AE⊥BC, ∴ AE⊥AD.

In Rt△ADE, DE=AD2+AE2 = (33)2+32=6.

∵△ADF∽△DEC, ∴ADDE=AFCD, ∴336=AF4,

∴AF= 23.

[Impression] A common method to prove that the products of line segments are equal is to convert the equation into a proportional expression, and then use the "three-point determination" to determine whether the triangles they are in are similar. If they are similar, the conclusion is established; if they are not similar, use the intermediate ratio Come "build a bridge".

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"Properties of Similar Triangles" PPT Courseware 3:

"Properties of Similar Triangles" PPT Courseware 3 Situation Introduction: Known: ABC∽A'B'C', according to the definition of similarity, what conclusions do we have? Viewed from corresponding sides: __________________ Viewed from corresponding angles: __________________ Two triangles..

"Properties of Similar Triangles" PPT Courseware 2:

"Properties of Similar Triangles" PPT Courseware 2 1. The ratios of the corresponding heights, the ratios of the corresponding midlines, and the ratios of the corresponding angle bisectors of similar triangles are all equal to ________. 2. The ratio of the perimeters of similar triangles is _________. 3． The ratio of the areas of similar triangles is equal to______..

"Determination of Similar Triangles" PPT Courseware 3:

"Determination of Similar Triangles" PPT Courseware 3 1. Three sides correspond to two proportional triangles ________. When using this judgment method to prove that two triangles are similar, pay attention to the corresponding relationship. Generally speaking, the opposite sides of equal angles are ________ sides. 2. Right triangle...

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Update Time: 2024-07-01

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