Question

a. the values of x and y are proportional. find the constant of proportionality (k) and complete the table.
x

x	9		5
y	36	16		8

a. set - up a proportion and solve to find the value of x.
(there are two triangles, the first triangle has a side of 25 yd and a side labeled x, the second triangle has a side of 15 yd and a side of 12 yd)
x =

1. what will be the new position of the given point (1, - 5) after reflection across the y - axis?

a) (1, 5)
b) (- 1, - 5)
c) (- 1, 5)
d) (1, - 5)

Explanation:

Part 1: Proportional Relationship (x and y)

Step 1: Find the constant of proportionality (k)

Since \( x \) and \( y \) are proportional, \( y = kx \). Using \( x = 9 \) and \( y = 36 \), we have \( 36 = k \times 9 \). Solving for \( k \), we get \( k=\frac{36}{9}=4 \).

Step 2: Find the missing x when \( y = 16 \)

Using \( y = kx \) and \( k = 4 \), \( 16 = 4x \). Solving for \( x \), we get \( x=\frac{16}{4}=4 \).

Step 3: Find the missing y when \( x = 5 \)

Using \( y = kx \) and \( k = 4 \), \( y = 4\times5 = 20 \).

Step 4: Find the missing x when \( y = 8 \)

Using \( y = kx \) and \( k = 4 \), \( 8 = 4x \). Solving for \( x \), we get \( x=\frac{8}{4}=2 \).

Step 1: Set up the proportion

Since the triangles are similar, the ratios of corresponding sides are equal. So, \( \frac{x}{12}=\frac{25}{15} \).

Step 2: Solve for \( x \)

Cross - multiply: \( 15x=12\times25 \). \( 15x = 300 \). Then \( x=\frac{300}{15}=20 \).

The rule for reflecting a point \( (x,y) \) across the \( y \) - axis is \( (x,y)\to(-x,y) \). For the point \( (1,-5) \), applying the rule, we change the sign of the \( x \) - coordinate. So the new point is \( (-1,-5) \).

Answer:

The constant of proportionality \( k = 4 \). The completed table is:

\( x \)	9	4	5	2
\( y \)	36	16	20	8

Part 2: Similar Triangles (Proportion)