Question

the larger black quadrilateral is a dilation of the smaller blue quadrilateral with point p as the center of dilation. select all that apply a the value of x is 8 cm. b the scale factor is \\(\frac{1}{2}\\) c the value of w is 106°. d the perimeter of the blue figure is 21.5 e the value of x is 6 cm. f the perimeter of the blue figure is 19.5

Explanation:

Step1: Identify Similar Figures and Scale Factor

The black quadrilateral is a dilation of the blue one, so they are similar. Find the scale factor using corresponding sides. For example, \( \frac{5}{2.5} = 2 \) or \( \frac{8}{4} = 2 \)? Wait, no—wait, blue has a side 4 cm, black has 8 cm? Wait, no, looking at the diagram: blue (smaller) has a side 4 cm, black (larger) has 8 cm? Wait, no, maybe the other way. Wait, the blue quadrilateral has a side of 2.5 cm, black has 5 cm: \( \frac{5}{2.5} = 2 \). Wait, but another side: blue has 4 cm, black has 8 cm? Wait, no, maybe the scale factor is \( \frac{1}{2} \)? Wait, no, let's check the options. Option B says scale factor \( \frac{1}{2} \)? Wait, no, let's re-examine.

Wait, the blue quadrilateral (smaller) and black (larger)? Wait, the problem says "the larger black quadrilateral is a dilation of the smaller blue quadrilateral". So black is larger, blue is smaller. So corresponding sides: blue has 2.5 cm, black has 5 cm: scale factor \( \frac{5}{2.5} = 2 \)? But option B says scale factor \( \frac{1}{2} \). Wait, maybe I got it reversed. Wait, maybe blue is larger? Wait, the diagram: blue has a side of 4 cm, black has 8 cm? No, the blue has 2.5 cm, black has 5 cm: 5/2.5=2. But let's check the options.

Wait, let's check each option:

A. x=8 cm: If scale factor is 2 (black is dilation of blue), then blue's side x? Wait, no, blue has a side of 4 cm, black has 8 cm? Wait, maybe the blue quadrilateral has a side of 4 cm, and black has 8 cm? Wait, the blue has a side of 2.5 cm, black has 5 cm: 5/2.5=2. So if blue has a side of 4 cm, black would have 8 cm. Wait, but option A says x=8 cm? Wait, no, maybe the blue quadrilateral's side corresponding to black's 12 cm? Wait, blue has a side of 6 cm? Wait, no, let's check option E: x=6 cm. Wait, maybe the scale factor is \( \frac{1}{2} \). Wait, let's take the blue quadrilateral: a side of 4 cm, black has 8 cm? No, maybe the blue is the larger one? Wait, the problem says "the larger black quadrilateral is a dilation of the smaller blue quadrilateral". So blue is smaller, black is larger. So scale factor is (black side)/(blue side). Let's find corresponding sides. Blue has a side of 2.5 cm, black has 5 cm: 5/2.5=2. Blue has 4 cm, black has 8 cm: 8/4=2. Blue has a side of x? Wait, no, the blue quadrilateral has a side labeled x, and black has 12 cm? Wait, maybe I messed up. Wait, let's look at the sides:

Blue quadrilateral (smaller) has sides: 2.5 cm, 4 cm, x, y.

Black quadrilateral (larger) has sides: 5 cm, 8 cm, 12 cm, 10 cm? Wait, no, the black has a side of 12 cm, 8 cm, 5 cm, and another. Wait, maybe the scale factor is \( \frac{1}{2} \), meaning black is a dilation with scale factor \( \frac{1}{2} \) of blue? No, that would make black smaller. Wait, the problem says "larger black" is dilation of "smaller blue", so scale factor >1. But option B says scale factor \( \frac{1}{2} \), which is <1. So maybe I got it reversed: blue is larger, black is smaller. So blue is larger, black is dilation (reduction) of blue. Then scale factor is \( \frac{1}{2} \), since 5/10? No, wait, blue has 5 cm, black has 2.5 cm? No, the diagram shows blue has 2.5 cm, black has 5 cm. Wait, this is confusing. Let's check the options:

Option E: x=6 cm. If scale factor is \( \frac{1}{2} \), then blue's side x corresponds to black's 12 cm: \( x = 12 \times \frac{1}{2} = 6 \) cm. Yes! So that works. So scale factor is \( \frac{1}{2} \) (black is a reduction of blue, so blue is larger, black is smaller). So:

- Scale factor: \( \frac{1}{2} \) (option B: "The scale factor \( = \frac{1}{2} \)") – correct.
- x: black has 12 cm, so blue's x is \( 12 \times \frac{1}{2} = 6 \) cm (option E: "The value of x is 6 cm" – correct).
- w: in similar figures, corresponding angles are equal. Black has an angle of 106°, so blue's w is 106°? Wait, no, the black has 106°, blue has w. Wait, the black quadrilateral has angles 100°, 106°, and others. In similar figures, corresponding angles are equal. So if black has 106°, then blue's w is 106°? Wait, option C: "The value of w is 106°" – correct? Wait, no, let's check the angles. The black quadrilateral (dilation of blue) – so corresponding angles are equal. So if black has 106°, then blue's w is 106° (option C correct).

Wait, let's re-express:

1. Scale Factor: Corresponding sides: blue (larger) has 5 cm, black (smaller) has 2.5 cm: \( \frac{2.5}{5} = \frac{1}{2} \). So scale factor is \( \frac{1}{2} \) (option B correct).

2. x: Black has 12 cm, so blue's x is \( 12 \times \frac{1}{2} = 6 \) cm (option E correct).

3. w: Corresponding angles are equal. Black has 106°, so blue's w is 106° (option C correct).

Now check perimeters:

Blue quadrilateral sides: x=6, y (let's see, black has 8 cm, so blue's y is \( 8 \times \frac{1}{2} = 4 \)? Wait, no, blue has 4 cm, black has 8 cm? Wait, no, blue has 4 cm, black has 8 cm? Then scale factor is 2. Wait, I'm getting confused. Let's list all sides:

Blue (smaller? Or larger?):

Sides: 2.5 cm, 4 cm, x, y.

Black (larger? Or smaller?):

Sides: 5 cm, 8 cm, 12 cm, 10 cm? Wait, the black has a side of 12 cm, 8 cm, 5 cm, and another. Wait, the blue has 2.5 cm (corresponding to black's 5 cm: 5/2.5=2), 4 cm (corresponding to black's 8 cm: 8/4=2), so scale factor is 2 (black is dilation of blue with scale factor 2). Then:

- x: blue has x, black has 12 cm: \( x \times 2 = 12 \) → \( x = 6 \) cm (option E correct).
- w: corresponding angles: black has 106°, so blue's w is 106°? Wait, no, in similar figures, corresponding angles are equal. So if black has 106°, blue has w=106° (option C correct).
- Scale factor: 2? But option B says \( \frac{1}{2} \). Contradiction. Wait, the problem says "the larger black quadrilateral is a dilation of the smaller blue quadrilateral". So black is larger, blue is smaller. So scale factor >1. So 5/2.5=2, 8/4=2, so scale factor 2. Then:

x: blue's side x, black's side 12 cm: \( x \times 2 = 12 \) → x=6 cm (option E correct).

w: black has 106°, so blue's w is 106°? Wait, no, the black quadrilateral's angle is 106°, and blue's angle w is corresponding, so w=106° (option C correct).

Scale factor: 2, but option B says \( \frac{1}{2} \). So option B is wrong. Wait, maybe the diagram is reversed. Let's check the sides again:

Blue quadrilateral (larger) has sides: 5 cm, 8 cm, 12 cm, 10 cm? No, the blue has 2.5 cm, 4 cm, x, y. Black has 5 cm, 8 cm, 12 cm, 10 cm. So 5/2.5=2, 8/4=2, 12/x=2 → x=6, 10/y=2 → y=5. Then perimeter of blue: 2.5 + 4 + 6 + 5 = 17.5? No, option D says perimeter of blue is 21.5, option F says black is 19.5. Wait, no, let's recalculate.

Wait, maybe I made a mistake. Let's start over.

1. **Similar Figures (Dilation) Properties**: Corresponding sides are proportional (scale factor), corresponding angles are equal.

2. **Identify Corresponding Sides**:
- Blue (smaller) has a side of 2.5 cm, black (larger) has 5 cm: \( \frac{5}{2.5} = 2 \) (scale factor \( k = 2 \)).
- Blue has 4 cm, black has 8 cm: \( \frac{8}{4} = 2 \) (consistent, \( k = 2 \)).
- Blue has side \( x \), black has 12 cm: \( \frac{12}{x} = 2 \) → \( x = 6 \) cm (option E: "The value of x is 6 cm" – correct).

3. **Corresponding Angles**:
- Black has an angle of 106°, so blue's corresponding angle \( w = 106^\circ \) (option C: "The value of w is 106°" – correct).

4. **Scale Factor**: \( k = 2 \), but option B says \( \frac{1}{2} \) – incorrect.

5. **Perimeter of Blue Quadrilateral**:
- Blue sides: 2.5 cm, 4 cm, \( x = 6 \) cm, and the fourth side: black has 10 cm (wait, black has a side of 10 cm? Wait, the diagram shows black has a side of 10 cm? No, the black quadrilateral has angles 100° and 106°, and sides 12 cm, 8 cm, 5 cm, 10 cm? Wait, blue has sides 2.5, 4, 6, and let's find the fourth side. Black's fourth side is 10 cm, so blue's is \( \frac{10}{2} = 5 \) cm. So blue perimeter: \( 2.5 + 4 + 6 + 5 = 17.5 \) cm. Option D says "The perimeter of the blue figure is 21.5" – incorrect. Option F says "The perimeter of the black figure is 19.5" – black perimeter: \( 12 + 8 + 5 + 10 = 35 \) cm? No, that's not 19.5. Wait, maybe I misread the sides.

Wait, the diagram: blue quadrilateral has a side of 2.5 cm, 4 cm, and black has 5 cm, 8 cm, 12 cm, and another side. Maybe the black quadrilateral has sides 12 cm, 8 cm, 5 cm, and 2.5×2=5? No, this is getting too confusing. Let's focus on the options that must be correct:

- **Option C**: \( w = 106^\circ \) (corresponding angles in similar figures are equal) – correct.
- **Option E**: \( x = 6 \) cm (scale factor 2, so \( 12 / 2 = 6 \)) – correct.
- Wait, but earlier I thought scale factor is 2, but maybe the problem has blue as larger. Wait, the problem says "the larger black quadrilateral is a dilation of the smaller blue quadrilateral" – so black is larger, blue is smaller. So scale factor \( k = 2 \), so blue's side \( x \) corresponds to black's 12 cm: \( x \times 2 = 12 \) → \( x = 6 \) (E correct). Blue's angle \( w \) corresponds to black's 106° (C correct).

Now, let's check option A: \( x = 8 \) cm – incorrect (we found x=6).

Option B: scale factor \( \frac{1}{2} \) – incorrect (scale factor is 2).

Option C: \( w = 106^\circ \) – correct.

Option D: blue perimeter 21.5 – let's calculate blue perimeter: 2.5 + 4 + 6 + (black's side / 2). Wait, black has a side of 10 cm? No, the black quadrilateral has a side of 10 cm? Wait, the diagram shows black has a side of 10 cm? No, the black quadrilateral has a side labeled 10 cm? Wait, no, the black quadrilateral has an angle of 100° and 106°, and sides 12, 8, 5, and another. Maybe the blue quadrilateral has a side of 4 cm, and black has 8 cm, blue has 2.5 cm, black has 5 cm, blue has x, black has 12 cm (so x=6), blue has a fourth side: black has 10 cm (so blue has 5 cm). Then blue perimeter: 2.5 + 4 + 6 + 5 = 17.5. Black perimeter: 5 + 8 + 12 + 10 = 35. 35 / 17.5 = 2 (scale factor), which matches. So option D (blue perimeter 21.5) is incorrect, option F (black perimeter 19.5) is incorrect.

So correct options are **C** and **E** (and maybe others, but let's confirm):

- **C**: \( w = 106^\circ \) (correct, corresponding angles).
- **E**: \( x = 6 \) cm (correct, scale factor 2, so 12 / 2 = 6).

Wait, but the problem says "Select all that apply". Let's recheck:

- **A**: x=8 – incorrect.
- **B**: scale factor \( \frac{1}{2} \) – incorrect (scale factor is 2).
- **C**: w=106° – correct.
- **D**: blue perimeter 21.5 – incorrect.
- **E**: x=6 – correct.
- **F**: black perimeter 19.5 – incorrect.

So the correct options are **C** and **E**. Wait, but maybe I missed something. Let's check the angle: the black quadrilateral has an angle of 106°, so blue's angle w (corresponding) is 106° – yes, because similar figures have equal corresponding angles. So C is correct. x=6: since black's side is 12, and scale factor is 2 (black is dilation of blue, so blue is smaller? No, wait, black is larger, so blue is smaller, so scale factor is 2 (black = blue × 2). So blue's x × 2 = black's 12 → x=6 – correct. So E is correct.

Answer:

C. The value of \( w \) is \( 106^\circ \), E. The value of \( x \) is \( 6 \) cm