Given that OA = ob = OC = a in tetrahedral o-abc, and the two are perpendicular, the radius r of the inscribed sphere of tetrahedral o-abc is calculated

Given that OA = ob = OC = a in tetrahedral o-abc, and the two are perpendicular, the radius r of the inscribed sphere of tetrahedral o-abc is calculated

Let s-abc be a regular tetrahedron with high sh, where h is the outer (inner, heavy and vertical) center of the triangle ABC on the bottom, connecting ah, making a vertical bisector of SA on the plane SAH, intersecting sh with O, then o is the inscribed (outer) center of the sphere,
Let the edge length be a, ah = a (√ 3 / 2) * (2 / 3) = a √ 3 / 3,
SH=√[a^2-(a√3/3)^2=a√6/3,
Let △ smo ∽ Sha, the radius of circumscribed sphere = R, the radius of inscribed sphere = R,
SM*SA=SO*SH,a^2/2=R*a√6/3,
R=a√6/4,
r=SH-SO=a√6/3-a√6/4=a√6/12.

Triangle cone o-abc, OA, ob, OC are vertical and equal,
If points P and Q are the moving points on BC and OA respectively, and 1 / 3bC ≤ BP ≤ 2 / 3bC and 1 / 3oa ≤ OQ ≤ 2 / 3oa are satisfied, then the value range of cosine value of angle between PQ and ob?

Let OA = ob = OC = 3, P (m, N, 0) & nbsp; Q (0,0, K) ∵ 1 / 3bC ≤ BP ≤ 2 / 3bC & nbsp; & nbsp; & nbsp;, 1 / 3oa ≤ OQ ≤ 2 / 3oa, ∵ & nbsp; 1 & lt; M & lt; 2,1 & lt; n {% 1}

In the triangular pyramid o-abc, if the three edges OA, OB and OC are perpendicular to each other, and OA = ob = OC, M is the midpoint of AB side, then the tangent value of the angle between OM and plane ABC is ()
A. 22B. 2C. 33D. 3

As shown in the figure: ∩ three edges OA, ob, OC are mutually perpendicular, and OA = ob = OC, ∩ AC = BC, OC ⊥ plane OAB. M is the midpoint of AB side, ∩ cm = m, ab ⊥ plane OCM, ∩ ab ⊂ plane ABC, ∩ plane OCM ⊥ plane ABC. It can be seen that OM is on the intersection line cm of two planes. ∩ OMC is the angle between OM and plane ABC. It is advisable to set om = 1, then OA = OC = 2 In △ OCM, Tan ∠ OMC = OCOM = 2