Analyze the diagram below and answer the question that follows

check the picture below.

so, notice those tickmarks, on the triangles KIH and KHM, the side KH is shared by both triangles, so is the same length in each triangle, and the angles on both triangles are the same.

At I vertex because they're linear angles, and the same at M vertex, linear angles, and at vertex K, because the taking out from 180° the other two angles in each triangle, the remaining angles, the ones at K in green, must be twins as well.

what the dickens does all that mean?

well, it means the triangles are congruent by ASA, as you see there, and since the triangles are congruent, they KI ≅ KM.

hello from mrbilldoesmath!

answer:   (7x + 6y) ( 49x^2 - 42xy + 36y^2)

discussion:

in general, the factorization of a^3 + b^3 is

a^3 + b^ 3 = (a + b) ( a^2 - ab + b^2)

in our case note that 343 = 7^3 and 216 = 6^3 so the equation becomes:

343x^3 + 216y^3 = (7x)^3 + (6y)^3

if you think of a (above) as 7x and b as 6y the factorization becomes

343x^3 + 216y^3 =

((7x + 6y) (   (7x)^2 - 7x*6y + (6y)^2)   =

(7x + 6y) ( 49x^2 - 42xy + 36y^2)

you,

mrb

(a + b) (a^2 - a b + b^2)