Chapter 11.7, Problem 3PSA

a.

To determine

To calculate: The ratio of the areas of $\Delta PQM$ and $\Delta PRM$

Answer to Problem 3PSA

The ratio of the areas of $\Delta PQM$ and $\Delta PRM$ is $1:1$

Explanation of Solution

Given: $\overline{PM}$ is a median

Calculation: Here, $A\left(\Delta PQM\right)=A\left(\Delta PRM\right)$

  → $\frac{A\left(\Delta PQM\right)}{A\left(\Delta PRM\right)}$ = $1$

  → $A\left(\Delta PQM\right):A\left(\Delta PRM\right)=1:1$

Conclusion: Hence, the ratio of the areas of $\Delta PQM$ and $\Delta PRM$ is $1:1$

b.

To determine

To calculate: The ratio of the areas of $\Delta PQM$ and $\Delta PQR$

Answer to Problem 3PSA

The ratio of the areas of $\Delta PQM$ and $\Delta PQR$ is $1:2$

Explanation of Solution

Given: $\overline{PM}$ is a median

Calculation: As, median divides the triangle into two equal halves.

  → $\frac{A\left(\Delta PQM\right)}{A\left(\Delta PQR\right)}=\frac{A\left(\Delta PQM\right)}{A\left(\Delta PQM\right)+A\left(\Delta PMR\right)}$

  → $\frac{1}{1+1}=\frac{1}{2}=1:2$

Conclusion: Hence, the ratio of the areas of $\Delta PQM$ and $\Delta PQR$ is $1:2$

c.

To determine

To calculate: The ratio of the sides QR and MR

Answer to Problem 3PSA

The ratio of the sides QR and MR is $2:1$

Explanation of Solution

Given: $\overline{PM}$ is a median

Calculation: Here, QM=MR

  $QR-MR=MR$

  $QR=2MR$

  $\frac{QR}{MR}=2$

  $QR:MR=2:1$

Conclusion: Hence, the ratio of the sides QR and MR is $2:1$