Question

a. the two flags are folded into triangles. the triangles are similar. find the lengths of \\(\overline{fh}\\) and \\(\overline{gh}\\).

triangles with labels: left triangle: vertices b, c, d; cd = 12 in, bd = 16 in. right triangle: vertices f, g, h; gh = 7x in, fh = 6x + 10 in.

the triangles are similar, so the corresponding sides are proportional.

\\(\frac{gh}{cd} = \frac{fh}{bd}\\)

\\(\frac{7x}{12} = \frac{6x + 10}{16}\\)

\\(12(6x + 10) = 16(7x)\\)

boxes filled with 72, 120, 112? wait, original has 12(6x + 10) = 16(7x) → 72x + 120 = 112x? wait, users image has some filled boxes, but ocr text here is the problem setup and equation steps.

then solving for x, then finding fh and gh:

fh = 6(□) + 10 = □ in.

gh = 7(□) = □ in.

Explanation:

Step1: Set up proportion

Since triangles are similar, \(\frac{GH}{CD}=\frac{FH}{BD}\). Substitute \(GH = 7x\), \(CD = 12\), \(FH=6x + 10\), \(BD = 16\): \(\frac{7x}{12}=\frac{6x + 10}{16}\)

Step2: Cross - multiply

Cross - multiply to get \(12(6x + 10)=16(7x)\)

Step3: Expand both sides

Expand: \(72x+120 = 112x\) (Wait, earlier mistake: \(12\times6x = 72x\), \(12\times10 = 120\), \(16\times7x=112x\), not 192x. Correcting that)

Step4: Solve for x

Subtract \(72x\) from both sides: \(120=112x - 72x\), \(120 = 40x\), so \(x = 3\)

Step5: Find FH

Substitute \(x = 3\) into \(FH=6x + 10\): \(FH=6\times3+10=18 + 10=28\)

Step6: Find GH

Substitute \(x = 3\) into \(GH = 7x\): \(GH=7\times3 = 21\)

Answer:

\(FH = 28\) in, \(GH = 21\) in