Question

1. what is the length of side gj?

options: 32.2, 15.5, 24.1, 28.2

Explanation:

Step1: Identify the figure type

The figure is a rectangle, so diagonals bisect each other. Thus, \( GK = KJ = 16 \), so the diagonal \( GJ \)? Wait, no, wait. Wait, in the rectangle \( GJIH \) (wait, the vertices are G, H, I, J? Wait, the sides: G to H, H to I, I to J, J to G? Wait, no, the rectangle has sides GJ and JI, with JI = 28. The diagonals are GH and... Wait, no, the diagonals are GI and HJ? Wait, no, the diagram shows diagonals intersecting at K, with GK = 16. Wait, maybe it's a rectangle, so the diagonals are equal and bisect each other. Wait, but we need to find GJ, which is a side, not a diagonal. Wait, maybe triangle GKJ? Wait, no, JI is 28, which is the length of the base. Wait, maybe the diagonals bisect each other, so \( GK = KI = 16 \)? Wait, no, the diagonal length would be \( 2 \times 16 = 32 \)? No, wait, maybe I misread. Wait, the side JI is 28, and the segment GK is 16. Wait, maybe triangle GJI? No, GJ is a vertical side, JI is horizontal (28), and the diagonal from G to I would be... Wait, no, the diagonals in a rectangle are equal and bisect each other. Wait, maybe the diagonals are GI and HJ, intersecting at K. So \( GK = KI = 16 \), so diagonal GI is 32? No, that can't be. Wait, maybe JI is 28, which is the length of the rectangle, and GJ is the height. Wait, maybe we can use the Pythagorean theorem. Wait, the diagonal from G to I: if GK is 16, then GI is 32? No, that would mean the diagonal is 32, but JI is 28. Then the height GJ would be \( \sqrt{32^2 - 28^2} \)? Wait, no, that would be if GI is the diagonal. Wait, let's recast:

Wait, the rectangle has length JI = 28, and the diagonal segment GK = 16, so the full diagonal GI is \( 2 \times 16 = 32 \)? Wait, no, maybe K is the midpoint, so GK = KI = 16, so diagonal GI = 32. Then, in the rectangle, the diagonal \( d = \sqrt{length^2 + height^2} \). So \( 32 = \sqrt{28^2 + GJ^2} \)? Wait, no, that would be \( GJ = \sqrt{32^2 - 28^2} \). Let's calculate that.

Step2: Apply Pythagorean theorem

\( GJ = \sqrt{32^2 - 28^2} \)
First, calculate \( 32^2 = 1024 \) and \( 28^2 = 784 \)
Then, \( 1024 - 784 = 240 \)
Wait, no, that's not right. Wait, maybe I got the diagonal wrong. Wait, maybe the diagonal is 32? No, 32 squared is 1024, 28 squared is 784, 1024 - 784 is 240, square root of 240 is about 15.5? No, that's not matching. Wait, maybe the diagonal is 32? No, wait, maybe the diagonal is 32, but that would make the height sqrt(32^2 -28^2)=sqrt(1024-784)=sqrt(240)≈15.5? But the options include 15.5. Wait, but let's check:

Wait, the problem is to find GJ. So if JI is 28 (length), and the diagonal GI is 32 (since GK=16, so GI=32), then GJ (height) is \( \sqrt{32^2 -28^2} = \sqrt{1024 - 784} = \sqrt{240} ≈ 15.5 \)? Wait, no, sqrt(240) is about 15.49, which is 15.5. So that's one of the options.

Wait, let's do the calculation:

\( 32^2 = 1024 \)

\( 28^2 = 784 \)

\( 1024 - 784 = 240 \)

\( \sqrt{240} ≈ 15.49 ≈ 15.5 \)

Step3: Confirm the calculation

So the length of GJ is approximately 15.5, which matches one of the options.

Answer:

15.5 (corresponding to the option with 15.5)