Let ABCD be a parallelogram. We have that M is the midpoint of AB and N is the midpoint of BC. The segments DM and DN intersect AC at P and Q, respectively. If AC = 15, what is QA?
Let ABCD be a parallelogram. We have that M is the midpoint of AB and N is the midpoint of BC.
The segments DM and DN intersect AC at P and Q, respectively.
If AC = 15, what is QA?
\(\text{Let $NC = \dfrac{AD}{2} $ } \)
intercept theorem:
\(\begin{array}{|rcll|} \hline \dfrac {QC}{NC} &=& \dfrac{QA}{AD} \quad & | \quad QC = AC - QA \\\\ \dfrac {AC - QA }{NC} &=& \dfrac{QA}{AD} \\\\ (AC - QA )\cdot AD &=& NC \cdot QA \quad & | \quad NC = \dfrac{AD}{2} \\\\ (AC - QA )\cdot AD &=& \dfrac{AD}{2} \cdot QA \quad & | \quad : AD \\\\ AC - QA &=& \dfrac{ QA}{2} \\\\ AC &=& QA + \dfrac{ QA}{2} \\\\ AC &=& \dfrac{3}{2}QA \\\\ \dfrac{3}{2}QA &=& AC \\\\ QA &=& \dfrac{2}{3}AC \quad & | \quad AC = 15 \\\\ QA &=& \dfrac{2}{3}\cdot 15 \\\\ QA &=& 2\cdot 5 \\\\ \mathbf{QA} & \mathbf{=} & \mathbf{10} \\ \hline \end{array}\)
Let ABCD be a parallelogram. We have that M is the midpoint of AB and N is the midpoint of BC.
The segments DM and DN intersect AC at P and Q, respectively.
If AC = 15, what is QA?
\(\text{Let $NC = \dfrac{AD}{2} $ } \)
intercept theorem:
\(\begin{array}{|rcll|} \hline \dfrac {QC}{NC} &=& \dfrac{QA}{AD} \quad & | \quad QC = AC - QA \\\\ \dfrac {AC - QA }{NC} &=& \dfrac{QA}{AD} \\\\ (AC - QA )\cdot AD &=& NC \cdot QA \quad & | \quad NC = \dfrac{AD}{2} \\\\ (AC - QA )\cdot AD &=& \dfrac{AD}{2} \cdot QA \quad & | \quad : AD \\\\ AC - QA &=& \dfrac{ QA}{2} \\\\ AC &=& QA + \dfrac{ QA}{2} \\\\ AC &=& \dfrac{3}{2}QA \\\\ \dfrac{3}{2}QA &=& AC \\\\ QA &=& \dfrac{2}{3}AC \quad & | \quad AC = 15 \\\\ QA &=& \dfrac{2}{3}\cdot 15 \\\\ QA &=& 2\cdot 5 \\\\ \mathbf{QA} & \mathbf{=} & \mathbf{10} \\ \hline \end{array}\)