Ask a New Question
Menu

draw △LMN where LM is a vertical line segment whose length is a multiple of 3 and MN is a horizontal line segment whose length is a multiple of 6 plot point q on LM such that LQ / Qm = 1 / 2. Similarly , plot point R on MN such that NR / RM = 1 / 2. Connect points Q and R to create QR. Then label Points L, M, N, Q and R with their coordinates

lasltly use the distance formula to show that LQ / QM = NR / RM.

1 answer

To illustrate the scenario described, I will use simple coordinate geometry to define the coordinates for the points and the relationships described. Here's how to construct the triangle \( \triangle LMN \) along with the points \( Q \) and \( R \).

  1. Define points L, M, and N:

    • Let \( L = (0, 0) \)
    • Let \( M = (0, 9) \) (This line segment \( LM \) has a length of 9, which is a multiple of 3)
    • Let \( N = (12, 0) \) (This line segment \( MN \) has a length of 12, which is a multiple of 6)

    Now the coordinates are:

    • \( L(0, 0) \)
    • \( M(0, 9) \)
    • \( N(12, 0) \)

  2. Plot point Q on LM such that \( \frac{LQ}{QM} = \frac{1}{2} \): Since \( Q \) divides \( LM \) in a \( 1:2 \) ratio, let \( LQ = x \) and \( QM = 2x \). The total length \( LM = LQ + QM = x + 2x = 3x = 9 \) implies \( x = 3 \). Thus, \( LQ = 3 \) and \( QM = 6 \).

    Therefore, the coordinates of \( Q \) are:

    • \( Q(0, 3) \)

  3. Plot point R on MN such that \( \frac{NR}{RM} = \frac{1}{2} \): Similarly, since \( R \) divides \( NM \) in a \( 1:2 \) ratio, let \( NR = y \) and \( RM = 2y \). The total length \( NM = NR + RM = y + 2y = 3y = 12 \) implies \( y = 4 \). Thus, \( NR = 4 \) and \( RM = 8 \).

    Therefore, the coordinates of \( R \) are:

    • \( R(8, 0) \)

  4. Now we have the coordinates:

    • \( L(0, 0) \)
    • \( M(0, 9) \)
    • \( N(12, 0) \)
    • \( Q(0, 3) \)
    • \( R(8, 0) \)

  5. Now, we'll find the distances \( LQ \), \( QM \), \( NR \), and \( RM \):

    • Distance \( LQ \): \[ LQ = \sqrt{(0 - 0)^2 + (3 - 0)^2} = \sqrt{0 + 9} = 3 \]

    • Distance \( QM \): \[ QM = \sqrt{(0 - 0)^2 + (9 - 3)^2} = \sqrt{0 + 36} = 6 \]

    • Distance \( NR \): \[ NR = \sqrt{(12 - 8)^2 + (0 - 0)^2} = \sqrt{4 + 0} = 4 \]

    • Distance \( RM \): \[ RM = \sqrt{(8 - 0)^2 + (0 - 9)^2} = \sqrt{64 + 81} = \sqrt{145} \]

  6. Comparing the ratios: We need to show that \( \frac{LQ}{QM} = \frac{NR}{RM} \).

    • Ratio \( \frac{LQ}{QM} = \frac{3}{6} = \frac{1}{2} \)
    • Ratio \( \frac{NR}{RM} = \frac{4}{8} = \frac{1}{2} \)

Thus, both ratios are equal, \( \frac{LQ}{QM} = \frac{NR}{RM} = \frac{1}{2} \), confirming the relationship.

This set of coordinates and calculations portrays the task you've outlined, showing the points and the specific ratios accordingly.