In right triangle PQR, the measure of angle Q is 90^\circ. If \sin(P) = \frac{5}{13} and t… — SAT

In right triangle $PQR$ , the measure of angle $Q$ is $90^\circ$ . If $\sin(P) = \frac{5}{13}$ and the length of side $QR$ is $15$ , what is the length of side $PQ$ ?

Cevap: 36

Cevap

The length of side

PQ

36

By definition,

\sin(P) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{QR}{PR}

. Given that

\sin(P) = \frac{5}{13}

and

QR = 15

, we set up the equation

\frac{5}{13} = \frac{15}{PR}

and solve for the hypotenuse

PR

, giving

PR = 39

. Using the Pythagorean theorem,

PQ^2 + QR^2 = PR^2

, we substitute the known values:

PQ^2 + 15^2 = 39^2 \implies PQ^2 + 225 = 1521 \implies PQ^2 = 1296

. Taking the square root of both sides gives

PQ = 36

. Alternatively, recognizing that the sides of the triangle form a

5

12

13

Pythagorean triple scaled by a factor of

3

(since

QR = 5 \times 3 = 15

and

PR = 13 \times 3 = 39

), the remaining leg

PQ

must be

12 \times 3 = 36

Adım Adım Çözüm

Set up the sine ratio for angle

P

to find the length of the hypotenuse

PR

PR = 39

Since

\sin(P)

is the ratio of the opposite side (

QR

) to the hypotenuse (

PR

), we can solve the equation

\frac{5}{13} = \frac{15}{PR}

to find that

PR = 39

Apply the Pythagorean theorem to solve for the length of side

PQ

PQ = 36

In right triangle

PQR

, the relationship between the sides is

PQ^2 + QR^2 = PR^2

. Substituting

QR = 15

and

PR = 39

gives

PQ^2 + 15^2 = 39^2

, which simplifies to

PQ^2 = 1296

, so

PQ = 36

Anahtar Kavram

Using trigonometric ratios to find side lengths of right triangles followed by the Pythagorean theorem.

Alternatif Yöntem

Recognize that the triangle's sides must be a multiple of the common

5

12

13

Pythagorean triple. Since the opposite side is

15

(

5 \times 3

) and the hypotenuse is

39

(

13 \times 3

), the scaling factor is

3

, meaning the adjacent side

PQ

12 \times 3 = 36

Tahmini Süre:1m 30s

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Adım Adım Çözüm

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