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ABCD∼QRST Find the missing side length, n. n=

ABCD∼QRST A B C D \sim Q R S T \newlineFind the missing side length, n n .\newlinen= \mathrm{n}=

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Full solution

Q. ABCD∼QRST A B C D \sim Q R S T \newlineFind the missing side length, n n .\newlinen= \mathrm{n}=
  1. Identify Similar Triangles: Since the triangles ABCDABCD and QRSTQRST are similar, corresponding side lengths are proportional. Let's assume we know the lengths of sides ABAB, CDCD, QRQR, and STST, and we need to find nn, the length of side RSRS.
  2. Set Up Proportion: Set up the proportion using the known side lengths. Suppose AB=8 cmAB = 8\,\text{cm}, CD=12 cmCD = 12\,\text{cm}, QR=16 cmQR = 16\,\text{cm}, and ST=nST = n. Since CDCD corresponds to STST, the proportion is CDST=ABQR\frac{CD}{ST} = \frac{AB}{QR}.
  3. Substitute Known Values: Substitute the known values into the proportion: 12n=816\frac{12}{n} = \frac{8}{16}.
  4. Simplify Equation: Simplify the right side of the equation: 816\frac{8}{16} reduces to 12\frac{1}{2}. So, 12n=12\frac{12}{n} = \frac{1}{2}.
  5. Cross-Multiply: Solve for nn by cross-multiplying: 12×2=n×112 \times 2 = n \times 1.
  6. Find Final Length: Simplify the equation: 24=n24 = n.

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