Math Help....

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1

<p>In the figure above, the circle has center P and radius r. Lines AB and AC are tangent to the circle.
If M is the midpoint of segment AC and the measure of ∠PMC equals the measure of ∠ MPC, what is the
length, in terms of r, of segment PA ?
If you need the picture of the figure, it’s in here: <a href=“College Board - SAT, AP, College Search and Admission Tools”>College Board - SAT, AP, College Search and Admission Tools;
Number 10 on the very last page. </p>

<p>(A) r + 1 (B) 2r (C) r rad2
(D) r rad3 (E) r rad5</p>

<p>Can someone please explain this to me? Thank you :slight_smile: </p>

2

<p>Thank you!</p>

3

<p>LEt me try to explain this purely mathematically:
∠PMC = ∠MPC => MC = PC (from tan 45 , or simple properties of traingles)
Thus MC = r .
Also, we know AM=MC, thus the length of AC is AM + MC= MC + MC = 2r.
PC²+AC²=AP² , or r²+(2r)² = AP² (pythagoras in APC)
or r²+4r²= AP²
or r²(1+5)= AP² (taking out r² as a common factor)
Thus, AP is r root 5 .</p>