Question

1. the polygons to the right are similar. find the value of each variable. x = y = (simplify your answers.) 7. the figures shown are similar. find the lengths of x, y, and z. the length of side x is (type an integer or a decimal.) the length of side y is (type an integer or a decimal.) the length of side z is (type an integer or a decimal.) 8. tzxy ~ fvxq. complete the statement below. \\(\frac{tz}{fv} = \frac{?}{xq}\\) \\(\frac{tz}{fv} = \frac{\\___}{xq}\\) 9. a 10 - ft vertical post casts a 20 - in shadow at the same time a nearby cell phone tower casts a 115 - ft shadow. how tall is the cell phone tower? the cell phone towers height is \\(\\___\\) ft. (simplify your answer.) 10. for the pair of similar triangles, find the appropriate value(s) of x. the appropriate value(s) of x is/are \\(\\___\\). (simplify your answer. use a comma to separate answers as needed.)

Explanation:

Step1: Find x via similar triangles

Since the triangles are similar, corresponding sides are proportional: $\frac{x}{12} = \frac{9}{15}$
$x = 12 \times \frac{9}{15} = 7.2$

Step2: Find y via similar triangles

Use proportionality of corresponding sides: $\frac{y}{9} = \frac{15}{9}$
$y = 9 \times \frac{15}{9} = 15$

Step3: Find x (problem7)

Similar figures have proportional sides: $\frac{x}{36} = \frac{16}{40}$
$x = 36 \times \frac{16}{40} = 14.4$

Step4: Find y (problem7)

Use proportionality: $\frac{y}{30} = \frac{16}{40}$
$y = 30 \times \frac{16}{40} = 12$

Step5: Find z (problem7)

Use proportionality: $\frac{z}{34} = \frac{16}{40}$
$z = 34 \times \frac{16}{40} = 13.6$

Step6: Complete similarity proportion

For $\triangle ZRY \sim \triangle FVX$, corresponding sides give $\frac{TZ}{FV} = \frac{RY}{VX}$

Step7: Find tower height

Set up proportion for shadows: $\frac{\text{Height of tower}}{115} = \frac{10}{20}$
$\text{Height of tower} = 115 \times \frac{10}{20} = 57.5$

Step8: Solve for x (problem10)

Similar right triangles have proportional sides: $\frac{2x-4}{24} = \frac{42}{x+8}$
Cross multiply: $(2x-4)(x+8) = 42 \times 24$
Expand: $2x^2 +16x -4x -32 = 1008$
Simplify: $2x^2 +12x -1040 = 0$ → $x^2 +6x -520 = 0$
Factor: $(x+26)(x-20)=0$
Valid positive solution: $x=20$

Answer:

1. $x=7.2$, $y=15$
2. The length of side x is 14.4; The length of side y is 12; The length of side z is 13.6
3. $\frac{TZ}{FV} = \frac{RY}{VX}$
4. The cell phone tower's height is 57.5 ft.
5. The appropriate value(s) of x is/are 20