Question

question 1 - 25
point x is located at (5.2, 0.6) and divides the line segment pq in the ratio 3:2. select two possible sets of endpoints for the line segment pq
□ p(4, 3), q(-6, 1)
□ p(4, 3), q(6, -1)
□ p(-1, 6), q(8, 3)
□ p(1, -6), q(6, 1)
□ p(1, 6), q(8, -3)

Explanation:

To determine which sets of endpoints \( P \) and \( Q \) have point \( X(5.2, 0.6) \) dividing \( PQ \) in the ratio \( 3:2 \), we use the section formula. For a point \( X(x, y) \) dividing the line segment joining \( P(x_1, y_1) \) and \( Q(x_2, y_2) \) in the ratio \( m:n \), the coordinates of \( X \) are given by:

$$ x = \frac{mx_2 + nx_1}{m + n}, \quad y = \frac{my_2 + ny_1}{m + n} $$

Here, \( m = 3 \) and \( n = 2 \), so:

$$ x = \frac{3x_2 + 2x_1}{5}, \quad y = \frac{3y_2 + 2y_1}{5} $$

We will test each option:

Option 1: \( P(4, 3) \), \( Q(-6, 1) \)

• Calculate \( x \)-coordinate:

$$ x = \frac{3(-6) + 2(4)}{5} = \frac{-18 + 8}{5} = \frac{-10}{5} = -2 $$

• Calculate \( y \)-coordinate:

$$ y = \frac{3(1) + 2(3)}{5} = \frac{3 + 6}{5} = \frac{9}{5} = 1.8 $$

• \( X(-2, 1.8)

eq (5.2, 0.6) \). So, this is not a valid set.

Option 2: \( P(4, 3) \), \( Q(6, -1) \)

• Calculate \( x \)-coordinate:

$$ x = \frac{3(6) + 2(4)}{5} = \frac{18 + 8}{5} = \frac{26}{5} = 5.2 $$

• Calculate \( y \)-coordinate:

$$ y = \frac{3(-1) + 2(3)}{5} = \frac{-3 + 6}{5} = \frac{3}{5} = 0.6 $$

• \( X(5.2, 0.6) \) matches. So, this is a valid set.

Option 3: \( P(-1, 6) \), \( Q(8, 3) \)

• Calculate \( x \)-coordinate:

$$ x = \frac{3(8) + 2(-1)}{5} = \frac{24 - 2}{5} = \frac{22}{5} = 4.4 $$

• Calculate \( y \)-coordinate:

$$ y = \frac{3(3) + 2(6)}{5} = \frac{9 + 12}{5} = \frac{21}{5} = 4.2 $$

• \( X(4.4, 4.2)

eq (5.2, 0.6) \). So, this is not a valid set.

Option 4: \( P(1, -6) \), \( Q(6, 1) \)

• Calculate \( x \)-coordinate:

$$ x = \frac{3(6) + 2(1)}{5} = \frac{18 + 2}{5} = \frac{20}{5} = 4 $$

• Calculate \( y \)-coordinate:

$$ y = \frac{3(1) + 2(-6)}{5} = \frac{3 - 12}{5} = \frac{-9}{5} = -1.8 $$

• \( X(4, -1.8)

eq (5.2, 0.6) \). So, this is not a valid set.

Option 5: \( P(1, 6) \), \( Q(8, -3) \)

• Calculate \( x \)-coordinate:

$$ x = \frac{3(8) + 2(1)}{5} = \frac{24 + 2}{5} = \frac{26}{5} = 5.2 $$

• Calculate \( y \)-coordinate:

$$ y = \frac{3(-3) + 2(6)}{5} = \frac{-9 + 12}{5} = \frac{3}{5} = 0.6 $$

• \( X(5.2, 0.6) \) matches. So, this is a valid set.

Valid Sets:

• \( P(4, 3) \), \( Q(6, -1) \)
• \( P(1, 6) \), \( Q(8, -3) \)

Answer:

B. \( P(4, 3), Q(6, -1) \)
E. \( P(1, 6), Q(8, -3) \)