Question

find ik.

triangle image with right angle at i, angle 51° at j, and jk = 9

write your answer as an integer or as a decimal rounded to the nearest tenth.

ik =

submit

Explanation:

Step1: Identify triangle type and trigonometric ratio

We have a right - triangle \( \triangle IJK \) with \( \angle I = 90^{\circ} \), \( \angle J=51^{\circ} \) and hypotenuse \( JK = 9 \). We want to find the length of \( IK \). In a right - triangle, the sine of an angle is defined as the ratio of the length of the opposite side to the hypotenuse. For \( \angle J \), the side opposite to it is \( IK \) and the hypotenuse is \( JK \). So, \( \sin(\angle J)=\frac{IK}{JK} \).

Step2: Substitute values and solve for \( IK \)

We know that \( \angle J = 51^{\circ} \) and \( JK = 9 \). From the sine formula \( \sin(51^{\circ})=\frac{IK}{9} \). To find \( IK \), we can multiply both sides of the equation by 9: \( IK=9\times\sin(51^{\circ}) \).
Using a calculator, \( \sin(51^{\circ})\approx0.7771 \). Then \( IK = 9\times0.7771\approx7.0 \) (rounded to the nearest tenth).

Answer:

\( 7.0 \)