Knowing that the triangle side AB 4 times the root number 3 and AC 2 are the root number 3 and A to

Solution: Do the midpoint e e of the ac edge, connect de;

then ae=ec= 3,de=1 2aba=2 3;

Look at the triangle ade, since ae= 3, de=2 3, ad=3de = ae +ad

So the triangle ade is a right triangle; and angle dae = 90 degrees;

Therefore, the triangular ADC is also a right-angled triangle, and the DC side is hypotenuse;

dc =ce +de =15, so dc = 15;

and bc=2dc=2 15.

Do the midpoint e e of the AC side and connect the de, then de is the median line of the triangle:

de=1 2ab=2 root number 3

ae = ec = root number 3

In Ade, ae=root3, de=2, root3, ad=3de2=ae2+ad2, so ade is a right-angled triangle, dae=90°

dc = root number (ad 2 + ac 2) = root number (3 2 + (2 root number 3) 2) = root number 21

bc = 2dc = 2 root number 21

According to the midline theorem, there is: ad 2 = ab 2 2 + ac 2 2-bc 2 4 substituting values are: 9 = 24 + 6-bc 2 4

Then there is: BC 2=84

i.e. bc = 2 21 (bc>0).

Root number 39 + root number 3) or (root number 39 - root number 3).

In the triangle ABC, ab = 4 times the root number 3, AC = 2 times the root number 3, ad is the midline on the side of BC, and the angle BAD=30°, find the length of BC.

Please give an accurate graph.

It is easiest to create a coordinate system and draw it in a coordinate system.

Because ab = 2 times the root number five ab squared is equal to 20, ac square is equal to 16, bc square = 4, according to the Pythagorean theorem, the triangle abc is a right triangle, c is a right angle, and ab is an hypotenuse; In the isosceles right-angled triangle ADB, the AB edge is also hypotenuse, so according to the theorem, if the AB edge is used as the diameter to make a circle, the points C and D are both on the edge of the circle. (Any point and diameter on the edge of the circle can form a right triangle) That is to say, the four points of abcd form a circle, so the line of cd is also a diameter, that is, cd=ab=2 times the root number five.

I don't know if I have a clear explanation, you can ask me again if you have any questions.

Method 1: Drawing method.

cd=√（9+9）=√18=

Method 2: Cosine Law.

a=45+ac=4 ad=√10=

cd ²=ac²+ad²-2ac*ad*∴cd=√18=

First of all, we use the inverse theorem of the Pythagorean theorem, and we can know that the triangle ACB is a right triangle.

Then lead the high ck perpendicular to ab to k, and de to h. Using the projective theorem of a right triangle, find ck, bk (i.e., dh). In this way, either cd or CD1 can be found.

Then, we draw the left and right rectangles using ab as the short side of the rectangle. Same method as above.

There are four types of cases. You can calculate it yourself. Just use the Pythagorean theorem.