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7 votes
7 votes
Geometry: Solve for x

Geometry: Solve for x-example-1
User Aleksejs Mjaliks
by
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1 Answer

19 votes
19 votes

Answer:


4√(3)

Explanation:

Since <PDA is right, the vertical angle, <BDC is right as well. Using sum of angles in a triangle, the angles in triangle BDC sum to 180, and for solving for <DCB, you get 30 degrees.

The line connecting the intersection of the tangents to the radius bisects the angle formed by the tangents; therefore, m<PCA is also 30 degrees.

A radius drawn to the point of tangency is perpendicular to the tangent, so <PAC is a right triangle. We can use sum of angles in a triangle for triangle PAC. Since <PCA is 30 and <PAC is right, <DPA is 60 degrees.

Now, we can look at triangle DPA, using sum of angles in a triangle, we can conclude triangle DPA is a 30-60-90 special right triangle. The leg opposite the 60 degree angle is √3 times the leg opposite the thirty degree angle. Since <DPA is 60 degrees and <PAD is 30:


DP√(3) = 6\\DP = (6)/(√(3) ) \\DP = (6)/(√(3) ) * (√(3) )/(√(3) ) \\DP = (6√(3) )/(3) \\DP = 2√(3)

Also, in a 30-60-90, the hypotenuse is twice the leg opposite the 30 degree angle, so x is 2 * 2√3. or
4√(3).

User Fjyaniez
by
2.8k points