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# Altitudes of triangle | PRMO 2017 | Question 17 Try this beautiful problem from the Pre-RMO, 2017 based on Altitudes of triangle.

## Altitude of Triangle - PRMO 2017

Suppose the altitudes of a triangle are 10, 12 and 15, find its semi perimeter.

• is 107
• $\frac{60}{\sqrt{7}}$
• is 840
• cannot be determined from the given information

### Key Concepts

Altitudes

Triangle

Semi-perimeter

Answer: is $\frac{60}{\sqrt{7}}$

PRMO, 2017, Question 17

Geometry Vol I to IV by Hall and Stevens

## Try with Hints

First hint

$h_a:h_b:h_c$=10:12:15

or, a:b:c=$\frac{1}{10} : \frac{1}{12} : \frac{1}{15}$=6:5:4

or, (a,b,c)=(6k,5k,4k)

or, 2s=15k

Second Hint

$\Delta=\sqrt{\frac{15k}{2}(\frac{15k}{2}-6k)(\frac{15k}{2}-5k)(\frac{15k}{2}-4k)}$

or, $\Delta=\frac{k^215\sqrt{7}}{4}$

Final Step

$h_{10}=10 =\frac{2k^2\sqrt{7}\frac{15}{4}}{6k}$

or, k=$\frac{8}{\sqrt{7}}$

or, s=$\frac{60}{\sqrt{7}}$

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Try this beautiful problem from the Pre-RMO, 2017 based on Altitudes of triangle.

## Altitude of Triangle - PRMO 2017

Suppose the altitudes of a triangle are 10, 12 and 15, find its semi perimeter.

• is 107
• $\frac{60}{\sqrt{7}}$
• is 840
• cannot be determined from the given information

### Key Concepts

Altitudes

Triangle

Semi-perimeter

Answer: is $\frac{60}{\sqrt{7}}$

PRMO, 2017, Question 17

Geometry Vol I to IV by Hall and Stevens

## Try with Hints

First hint

$h_a:h_b:h_c$=10:12:15

or, a:b:c=$\frac{1}{10} : \frac{1}{12} : \frac{1}{15}$=6:5:4

or, (a,b,c)=(6k,5k,4k)

or, 2s=15k

Second Hint

$\Delta=\sqrt{\frac{15k}{2}(\frac{15k}{2}-6k)(\frac{15k}{2}-5k)(\frac{15k}{2}-4k)}$

or, $\Delta=\frac{k^215\sqrt{7}}{4}$

Final Step

$h_{10}=10 =\frac{2k^2\sqrt{7}\frac{15}{4}}{6k}$

or, k=$\frac{8}{\sqrt{7}}$

or, s=$\frac{60}{\sqrt{7}}$

## Subscribe to Cheenta at Youtube

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