Question

1. find ad, ab, ae and ac. show work.

ad = ____ ab = ____
ae = ____ ac = ____

1. tony wants to find the distance across the lake.

use similar triangles to find this distance.

1. the triangles are similar. write and solve a proportion to solve for x.

Explanation:

Step1: Set up proportion for Q3

Since $DE \parallel BC$, $\triangle ADE \sim \triangle ABC$, so $\frac{AD}{AB} = \frac{AE}{AC}$. Substitute values: $\frac{x}{x+8} = \frac{x+4}{x+16}$

Step2: Cross-multiply to solve for x

$x(x+16) = (x+4)(x+8)$
$x^2 + 16x = x^2 + 12x + 32$
$16x - 12x = 32$
$4x = 32$
$x = 8$

Step3: Calculate lengths for Q3

$AD = x = 8$
$AB = x+8 = 8+8=16$
$AE = x+4 = 8+4=12$
$AC = 12+12=24$

Step4: Set up proportion for Q4

Let lake distance = $d$. Similar triangles give $\frac{d}{15} = \frac{16}{10}$

Step5: Solve for lake distance

$d = 15 \times \frac{16}{10} = 24$

Step6: Set up proportion for Q5

$\triangle MRS \sim \triangle MNP$, so $\frac{4}{4+x} = \frac{5}{5+x+1}$
$\frac{4}{x+4} = \frac{5}{x+6}$

Step7: Solve for x in Q5

$4(x+6) = 5(x+4)$
$4x +24 =5x +20$
$24-20=5x-4x$
$x=4$

Answer:

1. $AD = 8$, $AB = 16$, $AE = 12$, $AC = 24$
2. The distance across the lake is 24 m
3. $x = 4$