Help with Geometry!!

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Help with Geometry!!

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At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus. Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.

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In the picture AEC is right angled ar A,, P lies on AB, Q lies on BC, R lies on CA, and APQR is a square. The length of AB is 24 and the length of AC is 5. What is the length of AP? Write the answer as a common fraction
Let x be the side of the square... if AC = 5 then RC = (5 -x) and RQ = x and AB = 24 the triangles CRQ and ABC are similar by angle angle test.... I'll let you prove that... its easy then using ratios of corresponding sides... (5-x)/5 = x /24 or 120 - 24x = 5x 120 = 29x x = 120/29 units so AP = 120/29 units

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