Question

quadrilaterals whfp and kzns are shown. fpwh ~ kzns, fp = 3x - 0.4, wh = 4.3, pw = y + 1, hf = 3x, zn = 2y + 0.5, kz = 5x - 1.2, ns = 6.45, and sk = 3y - 0.6. solve for the variables, then determine the lengths of the sides.

Explanation:

Step1: Use the property of similar polygons (ratio of corresponding sides)

Since \(F P W H\sim K Z N S\), we have \(\frac{FP}{KZ}=\frac{WH}{NS}\) and \(\frac{PW}{ZN}=\frac{WH}{NS}\).
First, from \(\frac{FP}{KZ}=\frac{WH}{NS}\), substitute \(FP = 3x - 0.4\), \(KZ=5x - 1.2\), \(WH = 4.3\), and \(NS = 6.45\)

$$ \frac{3x - 0.4}{5x - 1.2}=\frac{4.3}{6.45} $$

Cross - multiply: \(6.45(3x - 0.4)=4.3(5x - 1.2)\)
Expand: \(19.35x-2.58 = 21.5x-5.16\)
Rearrange: \(21.5x-19.35x=5.16 - 2.58\)

$$ 2.15x=2.58 $$

Solve for \(x\): \(x=\frac{2.58}{2.15}=1.2\)

Step2: Use another ratio to solve for \(y\)

From \(\frac{PW}{ZN}=\frac{WH}{NS}\), substitute \(PW=y + 1\), \(ZN=2y + 0.5\), \(WH = 4.3\), and \(NS = 6.45\)

$$ \frac{y + 1}{2y+0.5}=\frac{4.3}{6.45} $$

Cross - multiply: \(6.45(y + 1)=4.3(2y + 0.5)\)
Expand: \(6.45y+6.45=8.6y + 2.15\)
Rearrange: \(8.6y-6.45y=6.45 - 2.15\)

$$ 2.15y=4.3 $$

Solve for \(y\): \(y = 2\)

Step3: Calculate the side lengths

For \(FP\): \(FP=3x - 0.4=3\times1.2-0.4=3.6 - 0.4 = 3.2\)
For \(KZ\): \(KZ=5x - 1.2=5\times1.2-1.2=6 - 1.2 = 4.8\)
For \(PW\): \(PW=y + 1=2 + 1=3\)
For \(ZN\): \(ZN=2y + 0.5=2\times2+0.5=4 + 0.5 = 4.5\)
For \(HF\): \(HF = 3x=3\times1.2 = 3.6\)
For \(SK\): \(SK=3y-0.6=3\times2-0.6=6 - 0.6 = 5.4\)

Answer:

\(x = 1.2\), \(y = 2\), \(FP = 3.2\), \(KZ = 4.8\), \(PW = 3\), \(ZN = 4.5\), \(HF = 3.6\), \(SK = 5.4\)