In the figure below, triangle ABC is a right triangle with the right angle at B. Point D l… — SAT

In the figure below, triangle $ABC$ is a right triangle with the right angle at $B$ . Point $D$ lies on side $AC$ such that segment $BD$ is perpendicular to side $AC$ . If $\cos(A) = \frac{4}{5}$ and the length of segment $AD$ is $16$ , what is the length of segment $CD$ ?

A
8
9Cevap
C
12
D
15

Cevap

The length of segment CD is 9.

The correct answer is the option indicating that the length of segment CD is 9. By applying the definition of cosine in right triangle ADB, the hypotenuse AB is found to be 20. The Pythagorean theorem then yields the length of the altitude BD as 12. Because triangle ADB is similar to triangle BDC, the ratio of the shorter leg to the longer leg is consistent between the triangles, giving the proportion CD/BD = BD/AD. Solving this proportion results in CD = 9.

Adım Adım Çözüm

Find the length of AB using the definition of cosine in right triangle ADB.

AB = 20

In right triangle ADB,

\cos(A) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{AD}{AB}

. Substituting the given values:

\frac{4}{5} = \frac{16}{AB}

, which simplifies to

AB = 20

Find the length of BD using the Pythagorean theorem in right triangle ADB.

BD = 12

Since ADB is a right triangle,

AD^2 + BD^2 = AB^2

. Substituting the known lengths:

16^2 + BD^2 = 20^2 \implies 256 + BD^2 = 400 \implies BD^2 = 144 \implies BD = 12

Use similar triangles to set up a proportion and solve for CD.

CD = 9

Triangles ADB and BDC are similar. Comparing the ratio of the shorter leg to the longer leg in both triangles gives

\frac{CD}{BD} = \frac{BD}{AD}

. Substituting the known values:

\frac{CD}{12} = \frac{12}{16} \implies CD = 12 \times \frac{3}{4} = 9

Anahtar Kavram

Using trigonometry and similar right triangles to find unknown segment lengths.

Soru

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Konular

Bu konudan daha fazla soru