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Nostrana [21]
3 years ago
7

The length of the hypotenuse of the right triangle is 25 cm, the length of one of the legs is 10 cm. What is the length of the s

egment, formed by the altitude to the hypotenuse that is adjacent to the other leg?

Mathematics
1 answer:
sergejj [24]3 years ago
3 0

Answer:

\large\boxed{AD=21\ and\ CD=2\sqrt{21}}

Step-by-step explanation:

Look at the picture.

Use the Pythagorean theorem for calculate the length of CA:

CA^2+CB^2=AB^2

We have

CA=x,\ CB=10,\ AB=25

Substitute:

x^2+10^2=25^2

x^2+100=625             subtract 100 from both sides

x^2=525\to x=\sqrt{525}\\\\x=\sqrt{25\cdot21}\\\\x=\sqrt{25}\cdot\sqrt{21}\\\\x=5\sqrt{21}\ cm

ΔADC and ΔACB are similar. Therefore the corresponding sides are in proportion:

\dfrac{AD}{AC}=\dfrac{AC}{AB}

We have:

AD=y,\ AC=5\sqrt{21},\ AB=25

Substitute:

\dfrac{y}{5\sqrt{21}}=\dfrac{5\sqrt{21}}{25}

\dfrac{y}{5\sqrt{21}}=\dfrac{\sqrt{21}}{5}              cross multiply

5y=5(\sqrt{21})^2            divide both sides by 5

y=21\ cm

If you mean the length of segment CD, then use the proportion:

\dfrac{CD}{AD}=\dfrac{CB}{CA}

Substitute:

\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

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