Lesson Finding the distanse from a point in 3D to a plane
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<H2>Finding the distanse from a point in 3D to a plane</H2> <H3>Problem 1</H3>Point P is at 10 cm distance from the vertices of a triangle with sides 4, 5, and 6 units long. Find the distance from point P to the plane of the triangle. <B>Solution</B> <pre> Consider the plane of the triangle and the sphere of the radius R = 10 cm with the center at the point P. This plane cuts the sphere and the triangle lies in/at the cutting section. Now what you need is to find the radius "r" of the circle circumscribed about the triangle. Then the distance from the point P to the plane is d = {{{sqrt(R^2 - r^2)}}}. (1) For the radius of the circle circumscribed about the triangle, there is the formula r = {{{abc/4A}}}, (2) where a, b, and c are the triangle's side dimensions and A is its area. See the lesson <A HREF=https://www.algebra.com/algebra/homework/Surface-area/Proof-of-the-formula-for-the-radius-of-the-circumscribed-circle.lesson>Proof of the formula for the radius of the circumscribed circle</A> in this site. To calculate the area, use the Heron's formula A = {{{sqrt(s*(s-a)*(s-b)*(s-c))}}} (where s is semi-perimeter) = {{{sqrt(7.5*(7.5-4)*(7.5-5)*(7.5-6))}}} = {{{sqrt(7.5*3.5*2.5*1.5)}}} = 9.922 cm^2 (approximately). Then according (2) r = {{{4*5*6/(4*9.922)}}} = 3.024 cm (approximately). Then the distance from P to the plane (which is the value under the question) is d = {{{sqrt(10^2-3.024^2)}}} = 9.532 cm. </pre> My other lessons on Miscellaneous advanced Geometry problems in this site are <TABLE> <TR> <TD> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/Triangle-with-the-side-ratio-456-has-a-smallest-angle-half-of-the-biggest-angle.lesson>Triangle with the sides ratio 4:5:6 has the smallest angle measure half of the biggest angle</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/Advanced-problem-on-equilateral-triangles-built-externally-on-sides-of-a-triangle-.lesson>Advanced problem on equilateral triangles built externally on sides of an arbitrary triangle</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/Advanced-problem-on-squares-built-externally-on-sides-of-a-triangle.lesson>Advanced problem on squares built externally on sides of an arbitrary triangle</A> - <A HREF=https://www.algebra.com/algebra/homework/word/geometry/Selected-problems-from-the-archive-on-the-area-of-plane-shapes.lesson>Selected problems from the archive on the area of plane shapes</A> - 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