Question

21 error analysis describe and correct the error in finding qu.

1. error analysis describe and correct the error in finding ad.

for question 21: there is a triangle with points r, t, u, s, q. rs = 5, rq = 6, st = 10, qu = x. the incorrect proportion is \\(\frac{rs}{qu} = \frac{rq}{st}\\) leading to \\(\frac{5}{x} = \frac{6}{10}\\), then 6x = 50, x = \\(8\frac{1}{3}\\).
for question 22: there is a triangle with points a, c, b, d. ad = x, dc = 14, ab = 10, bc = 16. the incorrect proportion is \\(\frac{ab}{bc} = \frac{cd}{ad}\\) leading to \\(\frac{10}{16} = \frac{14}{x}\\), then 10x = 224, x = 22.4.

Explanation:

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Problem 21

Step1: Identify correct proportion

Since $SQ \parallel TU$, the segments created are proportional: $\frac{RS}{ST} = \frac{RQ}{QU}$

Step2: Substitute given values

$\frac{5}{10} = \frac{6}{x}$

Step3: Cross-multiply to solve

$5x = 6 \times 10$
$5x = 60$
$x = \frac{60}{5} = 12$

Step1: Identify correct proportion

Since $BD$ is an angle bisector, by the Angle Bisector Theorem: $\frac{AB}{BC} = \frac{AD}{CD}$

Step2: Substitute given values

$\frac{10}{16} = \frac{x}{14}$

Step3: Cross-multiply to solve

$16x = 10 \times 14$
$16x = 140$
$x = \frac{140}{16} = 8.75$

Answer:

The error was setting up the proportion incorrectly (matching the wrong corresponding segments). The correct length of $QU$ is $12$.

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Problem 22