In the xy-coordinate plane, point A has coordinates (-9, 0) and point C has coordinates (0… — GRE General Test

In the $xy$ -coordinate plane, point $A$ has coordinates $(-9, 0)$ and point $C$ has coordinates $(0, 12)$ . Point $B$ lies on the positive $x$ -axis such that line segment $BD$ is perpendicular to segment $AC$ , with point $D$ lying on segment $AC$ . If the area of right triangle $ABD$ is $54$ , what is the length of segment $OB$ , where $O$ is the origin $(0,0)$ ?

Cevap: 6

Cevap

The length of segment

OB

6

The length of segment

OB

6

. Using the Pythagorean theorem on

\triangle AOC

, hypotenuse

AC = 15

, establishing a

3:4:5

side ratio for

\triangle AOC

. Because

\triangle ABD

shares acute angle

\angle A

with

\triangle AOC

and has a right angle at

D

\triangle ABD

is also a

3:4:5

right triangle with hypotenuse

AB

. Expressing the area

\frac{1}{2} \times \left(\frac{3}{5}AB\right) \times \left(\frac{4}{5}AB\right) = 54

yields

AB = 15

. Since

A

is at

(-9,0)

, point

B

is at

(6,0)

, making

OB = 6

Adım Adım Çözüm

Find the side lengths and hypotenuse of right triangle

AOC

Leg

AO = 9

, leg

OC = 12

, and by the Pythagorean theorem, hypotenuse

AC = \sqrt{9^2 + 12^2} = \sqrt{81 + 144} = \sqrt{225} = 15

The coordinates of

A(-9,0)

and

C(0,12)

form a right triangle at the origin

O(0,0)

Use angle similarity to determine the side ratio of right triangle

ABD

Triangle

ABD

is similar to triangle

AOC

because both contain a right angle and share

\angle DAO

. Thus, the sides of

\triangle ABD

maintain the ratio

AD : BD : AB = 3 : 4 : 5

Right triangles with a shared acute angle are similar.

Express legs

AD

and

BD

in terms of hypotenuse

AB

and set up the area equation.

AD = \frac{3}{5}AB

and

BD = \frac{4}{5}AB

. The area of

\triangle ABD = \frac{1}{2} \times AD \times BD = \frac{1}{2} \times \frac{3}{5}AB \times \frac{4}{5}AB = \frac{6}{25}AB^2

. Setting

\frac{6}{25}AB^2 = 54

yields

AB^2 = 225

, so

AB = 15

The area of a right triangle is half the product of its perpendicular legs.

Calculate the length of segment

OB

Since point

A

is at

(-9,0)

and

B

lies on the positive

x

-axis,

AB = x_B - (-9) = 15 \implies x_B = 6

. Therefore, the length of

OB

6

The distance from the origin

(0,0)

(6,0)

on the

x

-axis is equal to the

x

-coordinate

6

Anahtar Kavram

Applying Pythagorean triples (3-4-5 right triangle family) and similar right triangles in coordinate geometry.

Soru

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Konular

Bu konudan daha fazla soru