Sequences and Triangles question

Here is the question:



I would expect the answer to use a pythagoras calculation with a, a+d and a+2d, and I would expect the consecutive terms to be in descending order when they are reciprocals, so x is the biggest but its reciprocal would be the smallest.

I first tried the cosine rule using cos 90 as zero but the algebra becomes huge.

Is there a method I am missing?

Should I put everything in terms of a & d?

Surely any relationship between x, y & z is going to be concerning d?

No. If you use the a,d of the AP, you can eliminate them to get a relationship in x,y,z only. Or, follow my hint, and avoid a,d entirely.

No. If you use the a,d of the AP, you can eliminate them to get a relationship in x,y,z only. Or, follow my hint, and avoid a,d entirely.

Here's my work so far:

$\dfrac{1}{x}, \dfrac{z+x}{2xz}, \dfrac{1}{z}$

Putting this into pythagoras:

$(x+z)^2 = y^2 + (x-y+z)^2$