3 points on a straight line!

Geometry Level 3

The above shows two squares A B C D ABCD and E F G H EFGH with side lengths 10 and 6, respectively. Given that A , E , F A,E,F and B B lies on a straight line. And the straight line A E AE has a distance of 1.

If A I ⃗ = β E I ⃗ \vec{AI} = \beta \vec{EI} , what is the value of β \beta ?

3 5 \frac{3}{5} 1 2 \frac{1}{2} 5 3 \frac{5}{3} 2 3 \frac{2}{3}

Omar Sehlouli by Omar Sehlouli , 21, Morocco

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1 solution

Hana Wehbi
May 20, 2016

Triangles A D I a n d E H I ADI and\ EHI are similar triangles by A.A.A.

∠ I = ∠ I \angle I= \angle I Reflexive angle.

∠ D = ∠ H \angle D=\angle H Corresponding angles of parallel lines

∠ A = ∠ E \angle A=\angle E = 9 0 d e g 90 ^ {deg} Corresponding angles of parallel lines.

By taking the ratio of proptionality A I E I \frac{AI}{EI} = 10 6 \frac{10}{6} = 5 3 \frac{5}{3}

Keeping in mind two perpendiculars to the same line are parallel.

2 Helpful 0 Interesting 0 Brilliant 0 Confused

Great , note that A I E I = A D E H = D H D I = 5 3 \frac{AI}{EI} = \frac{AD}{EH} = \frac{DH}{DI} = \frac{5}{3}

Omar Sehlouli - 5 years ago

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You are right too.

Hana Wehbi - 5 years ago

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