Question

practice
example 1
use the figure at the right.

1. if ab = 6, bc = 4, and ae = 9, find ed.
2. if ab = 12, ac = 16, and ed = 5, find ae.

example 2
determine whether (overline{nr} parallel overline{pq}). justify your answer.

1. (pm = 18), (pn = 6), (qm = 24), and (rm = 16)
2. (qm = 31), (rm = 21), and (pm = 4pn)

example 3
find the value of x.

Explanation:

Step1: Set up proportion (Thales' Theorem)

$\frac{AB}{BC} = \frac{AE}{ED}$

Step2: Substitute given values (Q1)

$\frac{6}{4} = \frac{9}{ED}$

Step3: Solve for ED (cross-multiply)

$6 \times ED = 9 \times 4$
$ED = \frac{36}{6} = 6$

Step4: Set up proportion (Q2)

$\frac{AB}{BC} = \frac{AE}{ED}$
First find $BC = AC - AB = 16 - 12 = 4$

Step5: Substitute values (Q2)

$\frac{12}{4} = \frac{AE}{5}$

Step6: Solve for AE (cross-multiply)

$12 \times 5 = 4 \times AE$
$AE = \frac{60}{4} = 15$

Step7: Check proportional segments (Q3)

Check $\frac{PM}{PN} = \frac{QM}{RM}$
$\frac{18}{6} = 3$, $\frac{24}{16} = 1.5$

Step8: Solve for PN, check proportion (Q4)

$PM = 4PN \implies 4PN = PN + MN \implies 3PN = MN$
Check $\frac{PM}{PN} = \frac{QM}{RM}$: $\frac{4PN}{PN} = 4$, $\frac{31}{21} \approx 1.476$

Answer:

1. $ED = 6$
2. $AE = 15$
3. $\overline{NR}$ is not parallel to $\overline{PQ}$, because the segment ratios are not equal.
4. $\overline{NR}$ is not parallel to $\overline{PQ}$, because the segment ratios are not equal.