Question

self-assessment 2. find the value of x. abcd ~ qrst relate concepts when two similar polygons have a scale factor of k, what is the ratio of their perimeters? finding theore 8.1 pe if two po their peri correspor

Explanation:

Step1: Identify corresponding sides

Since \(ABCD \sim QRST\), the corresponding sides are proportional. Let's find the ratio of corresponding sides. For example, \(AB = 12\) corresponds to \(QR = 6\), \(AD = 10\) corresponds to \(QT = 5\), \(CD = 16\) corresponds to \(TS = 8\), and \(BC = x\) corresponds to \(RS = 4\).

Step2: Find the scale factor

Calculate the scale factor using a pair of corresponding sides. Let's use \(AB\) and \(QR\). The scale factor \(k=\frac{AB}{QR}=\frac{12}{6} = 2\). We can check with another pair: \(\frac{AD}{QT}=\frac{10}{5}=2\), \(\frac{CD}{TS}=\frac{16}{8}=2\). So the scale factor is 2.

Step3: Solve for \(x\)

Since \(BC\) corresponds to \(RS\), and the scale factor is 2, we have \(\frac{BC}{RS}=k\). Substituting the known values: \(\frac{x}{4}=2\). Multiply both sides by 4: \(x = 4\times2=8\)? Wait, no, wait. Wait, maybe I mixed up the correspondence. Wait, let's check the order of the similar polygons. \(ABCD \sim QRST\), so \(A\) corresponds to \(Q\), \(B\) to \(R\), \(C\) to \(S\), \(D\) to \(T\). So \(AB\) (12) corresponds to \(QR\) (6), \(BC\) (x) corresponds to \(RS\) (4), \(CD\) (16) corresponds to \(ST\) (8), \(DA\) (10) corresponds to \(TQ\) (5). So the ratio of \(ABCD\) to \(QRST\) is \(\frac{AB}{QR}=\frac{12}{6}=2\), so \(ABCD\) is twice as big as \(QRST\). Therefore, \(BC\) (x) should be \(RS\times2\)? Wait, no, \(RS\) is 4, so \(x = 4\times2 = 8\)? Wait, but let's check with \(CD\) and \(ST\): \(CD = 16\), \(ST = 8\), \(16/8 = 2\), so that's correct. \(DA = 10\), \(TQ = 5\), \(10/5 = 2\). \(AB = 12\), \(QR = 6\), \(12/6 = 2\). So yes, the scale factor from \(QRST\) to \(ABCD\) is 2. Therefore, \(BC\) (which is \(x\)) corresponds to \(RS\) (4), so \(x = 4\times2 = 8\)? Wait, but let's do it as proportions. Let's set up the proportion: \(\frac{AB}{QR}=\frac{BC}{RS}\). So \(\frac{12}{6}=\frac{x}{4}\). Cross - multiply: \(6x=12\times4\), \(6x = 48\), \(x=\frac{48}{6}=8\). Wait, but wait, maybe the correspondence is different? Wait, maybe \(ABCD\) and \(QRST\), so \(AB\) (12) corresponds to \(QR\) (6), \(BC\) (x) corresponds to \(RS\) (4), \(CD\) (16) corresponds to \(ST\) (8), \(DA\) (10) corresponds to \(TQ\) (5). So the ratio of \(AB\) to \(QR\) is 12/6 = 2, \(CD\) to \(ST\) is 16/8 = 2, \(DA\) to \(TQ\) is 10/5 = 2, so \(BC\) to \(RS\) should also be 2, so \(x/4 = 2\), so \(x = 8\). Wait, but let's check again. Wait, maybe I had the scale factor reversed. If \(QRST\) is similar to \(ABCD\), then the ratio of \(QRST\) to \(ABCD\) is 1/2. So \(RS\) is 4, so \(BC\) (x) is \(RS\times2\), so \(x = 8\). Yes, that makes sense.

Answer:

\(x = 8\)