Answer:
[tex]\large\boxed{AD=21\ and\ CD=2\sqrt{21}}[/tex]
Step-by-step explanation:
Look at the picture.
Use the Pythagorean theorem for calculate the length of CA:
[tex]CA^2+CB^2=AB^2[/tex]
We have
[tex]CA=x,\ CB=10,\ AB=25[/tex]
Substitute:
[tex]x^2+10^2=25^2[/tex]
[tex]x^2+100=625[/tex] subtract 100 from both sides
[tex]x^2=525\to x=\sqrt{525}\\\\x=\sqrt{25\cdot21}\\\\x=\sqrt{25}\cdot\sqrt{21}\\\\x=5\sqrt{21}\ cm[/tex]
ΔADC and ΔACB are similar. Therefore the corresponding sides are in proportion:
[tex]\dfrac{AD}{AC}=\dfrac{AC}{AB}[/tex]
We have:
[tex]AD=y,\ AC=5\sqrt{21},\ AB=25[/tex]
Substitute:
[tex]\dfrac{y}{5\sqrt{21}}=\dfrac{5\sqrt{21}}{25}[/tex]
[tex]\dfrac{y}{5\sqrt{21}}=\dfrac{\sqrt{21}}{5}[/tex] cross multiply
[tex]5y=5(\sqrt{21})^2[/tex] divide both sides by 5
[tex]y=21\ cm[/tex]
If you mean the length of segment CD, then use the proportion:
[tex]\dfrac{CD}{AD}=\dfrac{CB}{CA}[/tex]
Substitute:
[tex]\dfrac{CD}{21}=\dfrac{10}{5\sqrt{21}}\\\\\dfrac{CD}{21}=\dfrac{2}{\sqrt{21}}\qquad\text{cross multiply}\\\\CD\sqrt{21}=42\qquad\text{multiply both sides by}\ \sqrt{21}\\\\21CD=42\sqrt{21}\qquad\text{divide both sides by 21}\\\\CD=2\sqrt{21}\ cm[/tex]
