In right triangle ABC, the measure of angle C is 90^\circ, AC = 15, and BC = 20. Altitude … — SAT

In right triangle $ABC$ , the measure of angle $C$ is $90^\circ$ , $AC = 15$ , and $BC = 20$ . Altitude $CD$ is drawn to the hypotenuse $AB$ . From point $D$ , perpendicular line segments $DE$ and $DF$ are drawn to sides $AC$ and $BC$ , respectively, where $E$ lies on $AC$ and $F$ lies on $BC$ . What is the ratio of the area of triangle $CEF$ to the area of triangle $ABC$ ?

$\frac{144}{625}$ Answer
B
$\frac{12}{25}$
C
$\frac{9}{25}$
D
$\frac{16}{25}$

Answer

The ratio of the area of triangle

CEF

to the area of triangle

ABC

\frac{144}{625}

The correct answer is the fraction representing the ratio of the area of the smaller triangle to the larger triangle. This ratio can be computed by calculating the lengths of the legs of the right triangle

CEF

using similarity relationships, yielding

CE = 9.6

and

CF = 7.2

, which gives an area of

34.56

. Dividing this by the area of triangle

ABC

(

150

) yields the fraction

\frac{144}{625}

. Alternatively, the ratio is equal to the square of the ratio of the altitude to the hypotenuse,

(\frac{CD}{AB})^2 = (\frac{12}{25})^2 = \frac{144}{625}

Step-by-Step Solution

Use the Pythagorean theorem to calculate the hypotenuse

AB

of the right triangle

ABC

AB = \sqrt{AC^2 + BC^2} = \sqrt{15^2 + 20^2} = \sqrt{225 + 400} = \sqrt{625} = 25

Knowing the hypotenuse allows us to calculate the altitude and set up similarity ratios.

Calculate the area of triangle

ABC

\text{Area}(ABC) = \frac{1}{2} \times AC \times BC = \frac{1}{2} \times 15 \times 20 = 150

This establishes the base area to compare with the area of the smaller triangle.

Calculate the length of altitude

CD

by relating the area using base

AB

150 = \frac{1}{2} \times AB \times CD \Rightarrow 150 = \frac{1}{2} \times 25 \times CD \Rightarrow CD = 12

The altitude is a key dimension that relates the nested geometry to the main triangle.

Identify the properties of the quadrilateral

DECF

and the triangle

CEF

DECF

is a rectangle with diagonals

CD = EF = 12

, and

\triangle CEF

is a right triangle with legs

CE

and

CF

Since three angles are

90^\circ

(at

C

E

, and

F

), the figure is a rectangle, meaning its diagonals are congruent.

Apply similarity properties to find the lengths of

CE

and

CF

CE = 9.6

and

CF = 7.2

Using similar right triangles,

CE = AC - AE = 15 - 5.4 = 9.6

and

CF = BC - BF = 20 - 12.8 = 7.2

Calculate the area of right triangle

CEF

and find the ratio.

\text{Area}(CEF) = \frac{1}{2} \times 9.6 \times 7.2 = 34.56

. The ratio is

\frac{34.56}{150} = \frac{144}{625}

. Alternatively, the ratio is

\left(\frac{CD}{AB}\right)^2 = \left(\frac{12}{25}\right)^2 = \frac{144}{625}

Comparing the two areas gives the final required ratio.

Key Concept

Area ratios and side lengths in similar right triangles using perpendicular projections.

Alternative Method

Using trigonometric ratios, we can express the lengths in terms of

\theta = \angle A

. Since

CD = AB \sin\theta \cos\theta

, we have

CE = CD \sin\theta

and

CF = CD \cos\theta

. The area of triangle

CEF

\frac{1}{2} CE \cdot CF = \frac{1}{2} CD^2 \sin\theta \cos\theta = CD^2 \frac{\text{Area}(ABC)}{AB^2}

, which gives the area ratio as

\left(\frac{CD}{AB}\right)^2

. Since

AB = 25

and

CD = 12

, the ratio is

\left(\frac{12}{25}\right)^2 = \frac{144}{625}

Estimated Time:3m 0s

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Key Concept

Common Mistakes

Alternative Method

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