Mathematics for academic excellence
As shown in the figure, bac=90°, ab=4, ac=6
If you take BC as the edge and make the equilateral BCD downward, then AD=
Method 1: A triangular line is congruent
Make aec, make aec=60°, and take point f on the AB extension line, so that bfd=60°, make dg ab, from abc+ fbd=120°, fbd+ fdb=120°, get abc= fdb, and aec= bfd=60°, bc=bd get ace dbf, df=be=10, bf=ec=12, and gf=5 get bg=7, dg=5
In RT ADG, AD=14 is obtained from the Pythagorean theorem
Method 2: Construct congruence
There are the following three methods, the principle is the same, choose the left figure to illustrate.
In ad is the edge on the right side as an equilateral ade, connect CE, bdc= ade=60° to get bda= cde, and da=de, db=dc to get dba dce, so dce= dba, ce=ba, and abd+ acd=210°, so acd+ dce=210° to ace=150°, as eh ac in h, easy to know eh=2, ch=2
In RT AEH, the Pythagorean theorem yields AE=14, hence AD=14
Method 3: Direct hard calculation.
As shown in the figure, of course, this method is not recommended, the data setting is not very friendly, students can try it.
Method 4: Rotate + Similar
As shown in the figure, the construction of similar triangles gives a special angle of 150° in the same way, but this method is a good prescription to consider in different contexts. Students can use it as a method of understanding and accumulating spin + similarities.