Triangle question

What is the maximum perimeter of an obtuse-angled triangle with area $10cm^2$ and the longest side being $10cm$ ?

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

Markdown	Appears as
`italics` or `_italics_`	italics
`bold` or `__bold__`	bold
- bulleted - list	bulleted list
1. numbered 2. list	numbered list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1 paragraph 2	paragraph 1 paragraph 2
`[example link](https://brilliant.org)`	example link
`> This is a quote`	This is a quote
# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"	# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"

Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

@Pi Han Goh @Brandon Monsen @Otto Bretscher @Brian Charlesworth @Jon Haussmann any help?

Sumukh Bansal - 3 years, 6 months ago

Hint 1: area of a triangle, $\text{Area} = 10= \dfrac12 ab \sin C$ .

Hint 2: Cosine rule and double angle identities, $c^2 = 10^2 = a^2 + b^2 - 2ab \cos C$ , with $\cos C = 2\cos^2 \frac C2 - 1$ .

Hint 3: Connect the two 2 equations above.

Hint 4: Lagrange multipliers.

Pi Han Goh - 3 years, 6 months ago

@Pi Han Goh I understood the first 3 hints but what do you mean by Lagrange Multipliers.I am not able to understand and How to use them?

Sumukh Bansal - 3 years, 6 months ago

@Sumukh Bansal – You want to maximize the subject constraint $a + b + 10$ . You know the relationship between $a$ and $b$ . Lagrange multipliers is needed.

Pi Han Goh - 3 years, 6 months ago

@Pi Han Goh – Thanks.

Sumukh Bansal - 3 years, 6 months ago

@Pi Han Goh – @Pi Han Goh I am still not able to find the answer as I am not able to eliminate the variables.Could you help?

Sumukh Bansal - 3 years, 6 months ago

@Pi Han Goh By $\cos C = 2\cos^2 \frac C2 - 1$ you mean $\cos C = 2\cos^2 (\frac C2 - 1)$

Sumukh Bansal - 3 years, 6 months ago

@Sumukh Bansal – If I put $\cos C = 2\cos^2 \frac C2 - 1$ . Then how will I eliminate $\cos C$

Sumukh Bansal - 3 years, 6 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$