In rectangle ABCD, the length of side AB is 16 centimeters and the length of side BC is 12… — ACT

In rectangle $ABCD$ , the length of side $AB$ is $16$ centimeters and the length of side $BC$ is $12$ centimeters. Point $P$ lies on diagonal $AC$ such that segment $DP$ is perpendicular to $AC$ . What is the length, in centimeters, of segment $DP$ ?

Answer: 9.6 cm

Answer

The length of segment

DP

9.6

centimeters.

In rectangle

ABCD

, opposite sides are equal (

AD = BC = 12

cm and

DC = AB = 16

cm) and all interior angles are

90^\circ

. Right triangle

ADC

has legs

12

cm and

16

cm, making hypotenuse

AC = \sqrt{12^2 + 16^2} = 20

cm. Calculating the area of triangle

ADC

using the legs gives

\frac{1}{2} \times 12 \times 16 = 96

cm². Using hypotenuse

AC

as the base and perpendicular line segment

DP

as the altitude, the area is

\frac{1}{2} \times 20 \times DP = 10 \times DP

. Setting

10 \times DP = 96

yields

DP = 9.6

cm.

Step-by-Step Solution

Determine the length of diagonal

AC

Diagonal

AC = 20

Since

ABCD

is a rectangle, angle

ADC

is a right angle with legs

AD = 12

cm and

DC = 16

cm. By the Pythagorean theorem,

AC = \sqrt{12^2 + 16^2} = \sqrt{400} = 20

cm.

Calculate the area of right triangle

ADC

Area of triangle

ADC = 96

cm²

The area of a right triangle equals half the product of its legs:

\frac{1}{2} \times 12 \times 16 = 96

cm².

Solve for altitude

DP

Length

DP = 9.6

The area can also be expressed using hypotenuse

AC

as the base and

DP

as the altitude:

\text{Area} = \frac{1}{2} \times AC \times DP = \frac{1}{2} \times 20 \times DP = 10 \times DP

. Equating the two area expressions yields

10 \times DP = 96

, giving

DP = 9.6

cm.

Key Concept

Properties of Rectangles and Altitudes in Right Triangles

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