In right triangle RST, the measure of angle T is 90^\circ and the length of the hypotenuse… — SAT

In right triangle $RST$ , the measure of angle $T$ is $90^\circ$ and the length of the hypotenuse $RS$ is $10$ . If $\tan(R) + \tan(S) = \frac{5}{2}$ , what is the area of triangle $RST$ ?

A
10
20Answer
C
40
D
50

Answer

Let the lengths of the legs of right triangle

RST

RT = x

and

ST = y

. The tangent ratios are

\tan(R) = \frac{y}{x}

and

\tan(S) = \frac{x}{y}

. Substituting these into the given equation gives

\frac{y}{x} + \frac{x}{y} = \frac{5}{2}

, which simplifies to

\frac{x^2 + y^2}{xy} = \frac{5}{2}

. By the Pythagorean theorem,

x^2 + y^2 = 10^2 = 100

. Thus,

\frac{100}{xy} = \frac{5}{2}

, which solves to

xy = 40

. The area of the triangle is

\frac{1}{2}xy = \frac{1}{2}(40) = 20

Step-by-Step Solution

Express the tangent of angles

R

and

S

in terms of the leg lengths

RT

and

ST

\tan(R) = \frac{ST}{RT}

and

\tan(S) = \frac{RT}{ST}

By definition, the tangent of an acute angle in a right triangle is the ratio of the opposite leg to the adjacent leg.

Substitute these expressions into the given equation and simplify using the Pythagorean theorem.

\frac{ST}{RT} + \frac{RT}{ST} = \frac{ST^2 + RT^2}{RT \cdot ST} = \frac{100}{RT \cdot ST} = \frac{5}{2}

Finding a common denominator yields the sum of the squares of the legs in the numerator, which equals the square of the hypotenuse (

RS^2 = 10^2 = 100

Solve for the product of the legs

RT \cdot ST

and calculate the area of the triangle.

RT \cdot ST = 40

, so

\text{Area} = \frac{1}{2}(RT \cdot ST) = \frac{1}{2}(40) = 20

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

Key Concept

Using trigonometric ratios, complementary angles, and the Pythagorean theorem to calculate the area of a right triangle.

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