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發問:

a pyramid OPQR (trianglar base PQO, R at the top,PQ is a horigontal line, P is on the left, O is in the backside of the paper ),the sides OP,OQ,OR are the lengths x,y,z respectively,and the are mutually perpendicular to each other.i/ express cosPRQ in term of x,y,zii/ let s1,s2,s3 & s4 denote the areas of... 顯示更多 a pyramid OPQR (trianglar base PQO, R at the top,PQ is a horigontal line, P is on the left, O is in the backside of the paper ),the sides OP,OQ,OR are the lengths x,y,z respectively,and the are mutually perpendicular to each other. i/ express cosPRQ in term of x,y,z ii/ let s1,s2,s3 & s4 denote the areas of OPR,OPQ,OQR, and PQR respectively, show that s4^2=s1^2+s2^2+s3^2 (where s1^2 = s1 to the power of 2)

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最佳解答:

cosPRQ = (PR^2 + QR^2 - PQ^2) /2PR*QR = (x^2+z^2)+(y^2+z^2)-(x^2-y^2) /2sqrt(x^2+z^2)sqrt(y^2+z^2) =z^2/sqrt(x^2+z^2)sqrt(y^2+z^2) s4^2 =(1/2PR*QR*sinPRQ)^2 =1/4(PR^2*QR^2)(1-cos^2PRQ) =1/4(x^2+z^2)(y^2+z^2)[1- z^4/(x^2+z^2)(y^2+z^2)] =1/4[(x^2+z^2)(y^2+z^2)-z^4] =1/4[(xy)^2+(yz)^2+(zx)^2] =(xy/2)^2+(yz/2)^2+(zx/2)^2

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