Question

mixed exercises
identify the similar triangles. then find each measure.

1. ac
2. jl
3. eh
4. vt

Explanation:

Problem 9: Find $AC$

Step1: Identify similar triangles

$\triangle ABC \sim \triangle DEB$ (right triangles, shared $\angle E$)

Step2: Set up proportion

$\frac{AC}{DB} = \frac{BC}{EB}$
$\frac{x+1}{x+5} = \frac{12}{15}$

Step3: Simplify & solve for $x$

Simplify $\frac{12}{15}=\frac{4}{5}$, cross multiply:
$5(x+1)=4(x+5)$
$5x+5=4x+20$
$5x-4x=20-5$
$x=15$

Step4: Calculate $AC$

$AC=x+1=15+1=16$

---

Problem 10: Find $JL$

Step1: Identify similar triangles

$\triangle JKL \sim \triangle NML$ (vertical angles, parallel sides)

Step2: Set up proportion

$\frac{JL}{NL} = \frac{JK}{NM}$
$\frac{x+18}{x-3} = \frac{16}{4}$

Step3: Simplify & solve for $x$

Simplify $\frac{16}{4}=4$, cross multiply:
$x+18=4(x-3)$
$x+18=4x-12$
$18+12=4x-x$
$30=3x$
$x=10$

Step4: Calculate $JL$

$JL=x+18=10+18=28$

---

Problem 11: Find $EH$

Step1: Identify similar triangles

$\triangle EHG \sim \triangle EFD$ (parallel segments, shared $\angle E$)

Step2: Set up proportion

$\frac{EH}{EF} = \frac{HG}{FD}$
Let $EH = x+5$, so $EF=EH+HF=(x+5)+9=x+14$
$\frac{x+5}{x+14} = \frac{6}{12}$

Step3: Simplify & solve for $x$

Simplify $\frac{6}{12}=\frac{1}{2}$, cross multiply:
$2(x+5)=x+14$
$2x+10=x+14$
$2x-x=14-10$
$x=4$

Step4: Calculate $EH$

$EH=x+5=4+5=9$

---

Problem 12: Find $VT$

Step1: Identify similar triangles

$\triangle STU \sim \triangle VTU$? No, $\triangle RST \sim \triangle TVU$ (parallel segments, corresponding angles equal)

Step2: Set up proportion

$\frac{ST}{TU} = \frac{RS}{VU}$
$\frac{3x-3}{x+2} = \frac{14}{6}$

Step3: Simplify & solve for $x$

Simplify $\frac{14}{6}=\frac{7}{3}$, cross multiply:
$3(3x-3)=7(x+2)$
$9x-9=7x+14$
$9x-7x=14+9$
$2x=23$
$x=11.5$

Step4: Calculate $VT$

$VT=x+2=11.5+2=13.5$

Answer:

1. $AC = 16$
2. $JL = 28$
3. $EH = 9$
4. $VT = 13.5$