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Geometry Difficulty 5.2 AIME, harder Find the answer

In ABC\triangle A B C, BAC=5.25\angle B A C=5.25^{\circ}, ADA D is the angle bisector of BAC\angle B A C. Draw a perpendicular line from AA to DAD A intersecting the line BCB C at point MM. If BM=BA+ACB M=B A+A C, find the measures of ABC\angle A B C and ACB\angle A C B.

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

(ii) As follows,

Draw the perpendicular line from AA to DADA,
intersecting the extension of CBCB at point MM, extend BABA
to C1C_{1}, such that AC1AC_{1}
=AC = AC \text {. }

Connect AC1AC_{1},
C1CC_{1}C.
Then MAD=90\angle MAD=90^{\circ},
MAC=90+5.252=92.625 \therefore MAC=90^{\circ}+\frac{5.25^{\circ}}{2}=92.625^{\circ} \text {. }

Also, CAC1=1805.25=174.75\angle CAC_{1}=180^{\circ}-5.25^{\circ}=174.75^{\circ}.
Thus, MAC1=360174.75\angle MAC_{1}=360^{\circ}-174.75^{\circ}
92.625=92.625 -92.625^{\circ}=92.625^{\circ} \text {. }

Since MAC1=MAC\angle MAC_{1}=\angle MAC,
therefore MAC1MAC\triangle MAC_{1} \cong \triangle MAC.
Thus, MC1A=MCA=BCA\angle MC_{1}A=\angle MCA=\angle BCA.
Also, BC1=BA+AC1=AB+ACBC_{1}=BA+AC_{1}=AB+AC
=BM =BM \text {, }

Therefore, MC1A=C1MB\angle MC_{1}A=\angle C_{1}MB.
In MCC1\triangle MCC_{1},
3ACB+5.25=180,ACB=58.25. \begin{array}{l} 3 \angle ACB+5.25^{\circ}=180^{\circ}, \\ \angle ACB=58.25^{\circ} . \end{array}

Thus, ABC=18058.255.25\angle ABC=180^{\circ}-58.25^{\circ}-5.25^{\circ}
=116.5 =116.5^{\circ} \text {. }

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.