High School Math Solid Geometry Questions, High School Math Problems, Solid Geometry

The base is a regular triangle, and the projection of the vertex on the bottom is the triangular pyramid in the center of the base triangle.

Just do it by definition!

1, 4, it can be proved that the three side edges are equal. And the projection of the vertex on the bottom surface is the center of the bottom triangle (because it is a regular triangle, the four centers are one).

The bottom surface is an equilateral triangle, and the sides are isosceles triangles, and the triangular pyramid is a regular triangular pyramid.

This can be cited as a counter-example:

The regular triangle ABC is the base, and the straight line perpendicular to ABC is made through point A, and a point p. is taken on the straight lineConnect PB, PC

PB=PC triangle PBC is an isosceles triangle.

But it is clear that the triangular pyramid is not a regular triangular pyramid.

3.The problem can also give a counter-example:

Triangular pyramid P-ABC, the double-sided angle formed by the surface PBC and the bottom surface ABC is an obtuse angle, at this time, the projection of the P point on the surface ABC falls outside the triangle ABC shape, in addition, the height of the P point is AB, and the height of the AC will be on their extension line.

But at this time, the condition that the height of the side is equal is met!

Proposition 2 is not rigorous in the statement of isosceles triangles, and the title does not say which two sides are equal, maybe one side and one side of the equilateral triangle at the bottom are equal, but in this case, it is not a regular triangular pyramid.

Proposition 3, the side area is the same, and there is no problem of determining which side is the bottom edge, and the triangle area is divided by 2 by the bottom of the area, and the main problem is the determination of the bottom edge and the height here.

There is no picture, roughly speaking, I may still have to ponder.

I think 3 is also correct, if the 3 triangles are all based on the ground, then the height is equal, that is, the distance between the vertex and the 3 sides to the bottom is equal, let the length be d, and then set the distance between the vertex to the bottom is h, then the 3 dihedral angles have sin = h d, that is, the conclusion that the dihedral angles formed by the side and the bottom surface are equal.

1. The perpendicular line of the bottom surface is the center of the bottom surface.

This can be demonstrated by drawing! Draw patiently.

Make a straight line perpendicular to the plane at point E through point C, connect Be and AE Let Ce=X And because CB is 45 degrees to the plane, then Be=Ce=X then bc=root number 2*x In the same way, we get AE=X then we get AC=2x So ab=root number 6*x AD is high Find AD=2 3*root number 3*x sin@=x (2 3*root number3*X)=(root number 3) 2 So the size of the angle between AD and the plane is 60 degrees.

CE is perpendicular to m, the angle cae = 30 degrees, the angle cbe = 45 degrees, the angle cde is the angle sought, let ce = a, ca = 2a, cb = root number.

2a, cd = root number 3a 2, sin angle cde = ce cd = root number 3 2, angle cde = 60 degrees.

60°C1 is the bar of the auxiliary line and is perpendicular to the plane m

Then CC1=CA*SIN30°=CB*SIN45°The formula for a right-angled triangle is: AC2+CB2=AB 2, and because the area of the triangle can be obtained, AC*CB=CD*AB, so the angle is 60°

Question 2 7:5 EF is parallel to CB and E f is the midpoint of ACAB, then EF:CB=1:

2, then s aef:s acb=1:4, then saef:

S quadrilateral EFCB = 1:3 According to the formula of the area of the prism: (the area of the upper and bottom surface + the area of the lower bottom surface of the root number multiplied by the area of the bottom surface) * 1 3 * height then the area of the edge A b c -AEF is (4s + S + 2s) * 1 3 * h The area of the prism is:

4s*h then the remaining polygon area is: prism area - table area. It was calculated to be 7:5

A cross-section with an area of 5 and a radius of 5

A cross-section with an area of 8 and a radius of 8

Let the radius of the ball be r

Root number (R square - square of root 5) - root number (square of r - square of root 8) = 1r = 3

The answer is 15 sin60....There was originally a picture.,Hey.,The level is not enough.,Can't upload it.。。 Draw your own picture, draw a good picture and you can see: if the distance to the edge is x, sin60 = 15 x... Thus we get x,

Over d do DM vertical CE to mConnect the pm, pass d to make dn vertical pm to nIn the first question, it is easy to prove that PMD is the plane angle (45) of the dihedral angle P-EC-D, and the perpendicular bottom surface of Pd, so DM=PD=AD=1

Since thus be = root number 3 and ae = 2 - root number three. In the second question, the easy plane PDM is perpendicular to DC, so DCN is the line-surface angle, DN=DMSIN45 = root number 2 2, so the sine value is (root number 2) 4

I don't understand, I'll have to think about it