Originally Posted by
ticbol Cannot access the figure.
But, just for fun, basing only on the wordings of your posted question,
x = 36/37 = 0.973 unit long. --------answer.
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That is based on my assumption in the square MNPQ, where:
--AM is on the leg AB of the right triangle. (AB=3 units long; AM=x)
--NP is along the hypotenuse BC. (BC=5 units long)
--AQ is along the leg AC. (AC=4 units long)
Right triangle MAQ is similar to right triangle BAC, so, by proportion,
AM/AB = MQ/BC
Substituting values,
x/3 = s/5
s = 5(x/3) = 5x/3 unit long ----where s is a side of the square MNPQ.
Using proportions with similar triangles will get the x, but using trigonometry from here on is easier.
In rigth triangle BAC, let angle C = theta, then,
cos(theta) = AC/BC = 4/5 -----**
In right triangle MNB,
MN is perpendicular to BC,
BM is perpendicular to AC,
therefore, angle BMN is congruent to angle BCA, by angles whose sides are perpendicular each to each are congruent.
So, angle BMN = angle BCA = theta -----**
Then,
cos(theta) = MN/BM
where MN = s = 5x/3
So, substitutions,
4/5 = (5x/3)/BM
Cross multiply,
4(BM) = 5(5x/3)
BM = (25x/3)/4 = 25x/12 unit long.
But AB = BM +x = 3, so,
25x/12 +x = 3
Multiply both sides by 12,
25x +12x = 36
37x = 36
x = 36/37 = 0.973 unit long ----------answer.