Sometimes a geometry problem can be superimposed on the x-y plane to enable
an efficient solution. This is especially the case if there is a sequence
of points that are connected by a path in which each segment has a known slope,
and the endpoint is offset by a known amount in the x and y dimensions.

### Example 1

The given house has a curb roof. The upper rafters have a pitch of 1/4 and the lower rafters have a pitch of 1.
Find the lengths AB and CD. We are told the house is symmetrical about the
vertical line, CH, and we are given the total width (469 in) and height (244 in)
of the building.

A key to understanding this problem is knowing the architectural definition
of "pitch", which, in a symmetrical house like this one, is half the
slope. Now we know both the horizontal and vertical offset of point C from
point A, which is (469/2,244). And we can find a connected path from A to
C in which each segment has a known slope -- AB and BC.

At this point, we arbitrarily call segment AG "x". That makes
GB 2x, because the slope of AB is 2. Then, having traveled x inches
already, the distance from B to F is AH - AG, which is 469/2-x. Now, since
the slope of BC is 1/2, it follows that CF is half of BF.

Now, we have two expressions for the vertical distance CH: the sum of CF and
BG on the one hand, and CH on the other.

469/4 - x/2 + 2x = 244

x = 169/2

Now knowing that x=169/2, we can fill in the remaining values:

AG = x = 169/2

GB = 2x = 169

BF = 469/2-x = 150

CF = 469/4-x/2 = 75

Finally, the Pythagorean Theorem can be used to give us

AB = sqrt((169/2)^2+169^2) = 169/2*sqrt(5), or approximately 188.9477441

CD = BC = sqrt(75^2+150^2) = 75*sqrt(5), or approximately 167.7050983

### Internet references

Definition of "curb
roof" -- a roof with two or more pitches on each side of the ridge,
e.g. a mansard or gambrel roof

Definition of "roof
pitch" -- A roof's pitch is the measured vertical rise divided by the
measured horizontal span (not the run). And the span is defined as the
total distance between supports, so in practice, a roof's pitch is half it's
slope, because the denominator of pitch (the span) is twice the denominator of
the slope (the run).

### Related pages in this website

Geometry and Trigonometry

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Graeme McRae.