The lengths of HJ and KJ are:
HJ ≈ 13.6
KJ ≈ 16.7
A step-by-step explanation to determine the lengths of HJ and KJ:
Identify right triangles: Since triangles JKL and JHI are perpendicular with right angles at K and I, respectively, they are both right triangles.
Label triangles: Label the sides of each triangle:
Triangle JKL: JK = x, KL = 9, JL = 21
Triangle JHI: HI = 6, HJ = y, and JI = 16
Use Pythagorean Theorem: Since triangles JKL and JHI are right triangles, apply the Pythagorean Theorem to each triangle:
Triangle JKL: x^2 + 9^2 = 21^2
Triangle JHI: y^2 + 6^2 = 16^2
Solve for x:
Subtract 9^2 from both sides of the equation for triangle JKL: x^2 = 21^2 - 9^2
Take the square root of both sides: x = √(21^2 - 9^2)
Round to the nearest tenth: x ≈ 16.7
Solve for y:
Subtract 6^2 from both sides of the equation for triangle JHI: y^2 = 16^2 - 6^2
Take the square root of both sides: y = √(16^2 - 6^2)
Round to the nearest tenth: y ≈ 13.6
Therefore, the lengths of HJ and KJ are:
HJ ≈ 13.6
KJ ≈ 16.7