two parallel chords in a circle have lengths 10 and 14, and the distance between them is 6. The chord parallel to these chords and midway between them is of
length sqrt(a). Find the value of a.
+26367 two parallel chords in a circle have lengths 10 and 14, and the distance between them is 6.
The chord parallel to these chords and midway between them is of
length \(\sqrt{a}\). Find the value of a.
Phythagora's
\(\begin{array}{|lrcll|} \hline (1) & x^2+7^2 &=& r^2 \\ (2) & (3-x)^2 + \left(\dfrac{\sqrt{a}}{2}\right)^2 &=& r^2 \\ (3) & 5^2 + \Big(3+(3-x)\Big)^2 &=& r^2 \\ \hline \end{array} \)
\(\begin{array}{|lrcll|} \hline \mathbf{x=\ ?} \\ \hline (1)=(3): & r^2 = x^2+7^2 &=& 5^2 + \Big(3+(3-x)\Big)^2 \\ & x^2+7^2 &=& 5^2 + \Big(3+(3-x)\Big)^2 \\ & x^2+7^2 &=& 5^2 + (6-x)^2 \\ & x^2+7^2 &=& 5^2 + 6^2-12x+x^2 \\ & 7^2 &=& 5^2 + 6^2-12x \\ & 12x &=& 5^2 + 6^2 - 7^2 \\ & 12x &=& 12 \quad | \quad : 12 \\ & \mathbf{x}&=& \mathbf{1} \\ \hline \end{array}\)
\(\begin{array}{|lrcll|} \hline \mathbf{r^2=\ ?} \\ \hline (1): & r^2 &=& x^2+7^2 \quad | \quad \mathbf{x=1} \\ & r^2 &=& 1^2+7^2 \\ & \mathbf{r^2}&=& \mathbf{50} \\ \hline \end{array}\)
\(\begin{array}{|lrcll|} \hline \mathbf{a=\ ?} \\ \hline (2): & (3-x)^2 + \left(\dfrac{\sqrt{a}}{2}\right)^2 &=& r^2 \\ & (3-x)^2 + \dfrac{a}{4} &=& r^2 \quad | \quad \mathbf{x=1},\ \mathbf{r^2=50} \\ & (3-1)^2 + \dfrac{a}{4} &=& 50 \\ & 2^2 + \dfrac{a}{4} &=& 50 \\ & 4 + \dfrac{a}{4} &=& 50 \quad | \quad -4 \\ & \dfrac{a}{4} &=& 46 \quad | \quad * 4 \\ & \mathbf{a}&=& \mathbf{184} \\ \hline \end{array}\)
+26367 two parallel chords in a circle have lengths 10 and 14, and the distance between them is 6.
The chord parallel to these chords and midway between them is of
length \(\sqrt{a}\). Find the value of a.
Phythagora's
\(\begin{array}{|lrcll|} \hline (1) & x^2+7^2 &=& r^2 \\ (2) & (3-x)^2 + \left(\dfrac{\sqrt{a}}{2}\right)^2 &=& r^2 \\ (3) & 5^2 + \Big(3+(3-x)\Big)^2 &=& r^2 \\ \hline \end{array} \)
\(\begin{array}{|lrcll|} \hline \mathbf{x=\ ?} \\ \hline (1)=(3): & r^2 = x^2+7^2 &=& 5^2 + \Big(3+(3-x)\Big)^2 \\ & x^2+7^2 &=& 5^2 + \Big(3+(3-x)\Big)^2 \\ & x^2+7^2 &=& 5^2 + (6-x)^2 \\ & x^2+7^2 &=& 5^2 + 6^2-12x+x^2 \\ & 7^2 &=& 5^2 + 6^2-12x \\ & 12x &=& 5^2 + 6^2 - 7^2 \\ & 12x &=& 12 \quad | \quad : 12 \\ & \mathbf{x}&=& \mathbf{1} \\ \hline \end{array}\)
\(\begin{array}{|lrcll|} \hline \mathbf{r^2=\ ?} \\ \hline (1): & r^2 &=& x^2+7^2 \quad | \quad \mathbf{x=1} \\ & r^2 &=& 1^2+7^2 \\ & \mathbf{r^2}&=& \mathbf{50} \\ \hline \end{array}\)
\(\begin{array}{|lrcll|} \hline \mathbf{a=\ ?} \\ \hline (2): & (3-x)^2 + \left(\dfrac{\sqrt{a}}{2}\right)^2 &=& r^2 \\ & (3-x)^2 + \dfrac{a}{4} &=& r^2 \quad | \quad \mathbf{x=1},\ \mathbf{r^2=50} \\ & (3-1)^2 + \dfrac{a}{4} &=& 50 \\ & 2^2 + \dfrac{a}{4} &=& 50 \\ & 4 + \dfrac{a}{4} &=& 50 \quad | \quad -4 \\ & \dfrac{a}{4} &=& 46 \quad | \quad * 4 \\ & \mathbf{a}&=& \mathbf{184} \\ \hline \end{array}\)
