Question

En el triángulo $CAT,$ tenemos $\angle ACT =\angle ATC$ y $\angle CAT = 36^\circ.$ Si $\overline{TR}$ biseca $\angle ATC,$ entonces $\angle CRT =$

In triangle $CAT,$ we have $\angle ACT =\angle ATC$ and $\angle CAT = 36^\circ.$ If $\overline{TR}$ bisects $\angle ATC,$ then $\angle CRT =$

$16^\circ$

$51^\circ$

$72^\circ$

$90^\circ$

$108^\circ$

Solución:

Obtenemos $\begin{aligned} \angle CAT + \angle ATC \\ \quad {}+ \angle ACT &= 180 \\ 36 + 2 \cdot \angle ATC &= 180 \\ \angle ATC &= 72^{\circ}. \end{aligned}$

Debido a la bisección, también sabemos que $\angle RTC = 72 \div 2 = 36^{\circ}.$

Finalmente, vemos que $\begin{aligned} \angle RTC + \angle TCR \\ \quad {}+ \angle CRT &= 180 \\ 36 + 72 + \angle CRT &= 180 \\ \angle CRT &= 72^{\circ}. \end{aligned}$

Así, C es la respuesta correcta.

We get $\begin{aligned} \angle CAT + \angle ATC \\ \quad {}+ \angle ACT &= 180 \\ 36 + 2 \cdot \angle ATC &= 180 \\ \angle ATC &= 72^{\circ}. \end{aligned}$

Due to bisection, we also know that $\angle RTC = 72 \div 2 = 36^{\circ}.$

Finally, we see that $\begin{aligned} \angle RTC + \angle TCR \\ \quad {}+ \angle CRT &= 180 \\ 36 + 72 + \angle CRT &= 180 \\ \angle CRT &= 72^{\circ}. \end{aligned}$

Thus, C is the correct answer.

2000 AMC 8 Problema 13

Solución:

El Problema 13 en otros años