In right triangle ABC with \angle B = 90^\circ and \angle A = 30^\circ, the hypotenuse AC … — ACT

In right triangle $ABC$ with $\angle B = 90^\circ$ and $\angle A = 30^\circ$ , the hypotenuse $AC$ has a length of $12$ centimeters. An altitude $BD$ is drawn from vertex $B$ to hypotenuse $AC$ . From point $D$ , a perpendicular segment $DE$ is drawn to side $AB$ , with point $E$ lying on $AB$ . What is the length, in centimeters, of segment $EC$ ?

A
$3\sqrt{7}$
B
$\frac{3\sqrt{22}}{2}$
$\frac{3\sqrt{19}}{2}$ Cevap
D
$\frac{3\sqrt{13}}{2}$
E
$\sqrt{39}$

Cevap

The length of segment EC is \frac{3\sqrt{19}}{2} centimeters.

To find the length of segment EC, we can construct the right triangle EBC with a right angle at B. By using the properties of 30-60-90 right triangles, we determine the side lengths of the triangles in the figure: first finding BC = 6 and AB = 6\sqrt{3} in triangle ABC; then finding AD = 9 in triangle ABD; then finding AE = \frac{9\sqrt{3}}{2} in triangle ADE; and finally finding EB = AB - AE = \frac{3\sqrt{3}}{2}. Applying the Pythagorean theorem to right triangle EBC yields EC = \sqrt{(\frac{3\sqrt{3}}{2})^2 + 6^2} = \frac{3\sqrt{19}}{2}.

Adım Adım Çözüm

Determine the side lengths of the main right triangle ABC. Since angle A = 30 degrees and angle B = 90 degrees, triangle ABC is a 30-60-90 right triangle. With hypotenuse AC = 12, the leg opposite the 30-degree angle is BC = \frac{12}{2} = 6, and the leg opposite the 60-degree angle is AB = 6\sqrt{3}.

BC = 6 and AB = 6\sqrt{3}

Knowing the side lengths of triangle ABC is necessary to find the dimensions of the smaller inscribed triangles.

Find the length of segment AD in right triangle ABD. Altitude BD is perpendicular to AC, making triangle ABD a right triangle with right angle ADB. Since angle A = 30 degrees, triangle ABD is also a 30-60-90 right triangle with hypotenuse AB = 6\sqrt{3}. The side adjacent to the 30-degree angle, AD, is given by AB \times \cos(30^\circ) = 6\sqrt{3} \times \frac{\sqrt{3}}{2} = 9.

AD = 9

Determining AD allows us to analyze the smaller right triangle ADE built on it.

Find the lengths of segments AE and EB. In right triangle ADE (where DE is perpendicular to AB), the hypotenuse is AD = 9 and angle A = 30 degrees. The side adjacent to the 30-degree angle is AE = AD \times \cos(30^\circ) = 9 \times \frac{\sqrt{3}}{2} = \frac{9\sqrt{3}}{2}. Segment EB is then found by subtracting AE from AB: EB = AB - AE = 6\sqrt{3} - \frac{9\sqrt{3}}{2} = \frac{3\sqrt{3}}{2}.

EB = \frac{3\sqrt{3}}{2}

We need the length of segment EB to apply the Pythagorean theorem in the final right triangle EBC.

Apply the Pythagorean theorem to right triangle EBC. Since line segment AB is perpendicular to BC, angle EBC is a right angle. In right triangle EBC, the legs are EB = \frac{3\sqrt{3}}{2} and BC = 6. The hypotenuse EC is calculated as EC = \sqrt{EB^2 + BC^2} = \sqrt{(\frac{3\sqrt{3}}{2})^2 + 6^2} = \sqrt{\frac{27}{4} + 36} = \sqrt{\frac{171}{4}} = \frac{3\sqrt{19}}{2}.

EC = \frac{3\sqrt{19}}{2}

Applying the Pythagorean theorem to the legs EB and BC gives the length of the hypotenuse EC.

Anahtar Kavram

Applying 30-60-90 right triangle properties and the Pythagorean theorem across multiple connected geometric figures.

Tahmini Süre:3m 0s

Soru

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Konular

Bu konudan daha fazla soru