Staircase Heights

Geometry Level 1

An architect is trying to design a short staircase. His plan is shown in the diagram below, where all the vertical and horizontal lines meet at right angles.

In the figure below, $AB = BC$ and $\angle ABC=90^\circ.$ What is the height of the third step $($ marked $HI)?$

Note: No variables or calculations are needed to solve this problem.

\frac{1}{8}

\frac{1}{6}

\frac{1}{3}

3

3 solutions

Syed Hamza Khalid
Nov 25, 2017

Simply: $\large \frac{1}{2} ÷ 4 = \frac{1}{8}$

Believe it or not, I solved this in 2 seconds

how does this work?

Matt Watta - 3 years, 6 months ago

As there are 4 vertical heights, each increased by 0.5 (less than each) . I will surely explain more after my a week since I am having exams! :((

Syed Hamza Khalid - 3 years, 6 months ago

Vance Noonan
Dec 1, 2017

Because all of the blue legs are either horizontal or vertical, for all of the triangles with their hypotenuse along the DC line, the vertical line is $\frac{1}{2}$ the horizontal line. Since AB=BC, line AC is a 45 degree angle. Therefore, for all triangles with their hypotenuse along the AC line, the horizontal line equals the vertical. So:

AD = 1
AE = $\frac{1}{2}$ $\times$ AD = $\frac{1}{2}$ = EF
FG = $\frac{1}{2}$ $\times$ EF = $\frac{1}{4}$ = GH
HI = $\frac{1}{2}$ $\times$ GH = $\frac{1}{8}$

Vincent Miller Moral
Dec 1, 2017

HINT: Tri. EAD is a 30-60-90 triangle while Tri. AEF is isosceles right.

Staircase Heights

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