Length of a Side

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In triangle $ABC$ , $\angle B=30^{\circ}$ , $\angle C=90^{\circ}$ , and $D$ is a point on side $BC$ such that $\overline{BD}=56$ and $\angle ADC=45^{\circ}.$ The length of side $AC$ can be expressed as $p\sqrt{3}+p.$ What is the value of $p?$

30

31

29

28

1 solution

Finn Hulse
Feb 11, 2014

We see that that CDA is 45 degrees, and DCA is 90, therefore DAC is 45. This means that DAC is a 45-45-90 triangle. Let's name side CA x, seeing as it's what we're solving for. That means that CD is also x (45-45-90 triangles have both legs equal). Zooming out, we see that the whole triangle (ABC) is a 30-60-90 triangle. Therefore the hypotenuse is 2x, and BC is the square root of 3 times x. Side BC is also 56 + x. So we set up the equation: $x\sqrt{3}=x+56$ . Solving, find that p is 28, the answer.

why is the hypotenuse equal to 2x?

DPK - 7 years, 3 months ago

Length of a Side

1 solution

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