Generalisation of "Perimeter from Altitudes"

If ΔABC\Delta ABC has altitudes AD=pAD = p, BE=qBE = q, CF=rCF = r, then we can find the perimeter in terms of p,q,rp, q, r.

Since the area of ΔABC\Delta ABC is constant, ABr=ACq=BCpAB \cdot r = AC \cdot q = BC \cdot p. If we let another similar triangle ΔABC\Delta A'B'C' satisfy AB=pq,AC=pr,BC=qrA'B' = pq, A'C' = pr, B'C' = qr, then we can ensure this equality holds true.

However, the length of the altitudes will not be p,q,rp, q, r, but scaled by a constant kk. Now let AF=xA'F' = x, so BF=pqxB'F' = pq - x. Computing CFC'F' two different ways using Pythagoras, (pr)2x2=(qr)2(pqx)2(pr)^2 - x^2 = (qr)^2 - (pq -x)^2 , which simplifies to (pr)2x2=(qr)2((pq)22pqx+x2)(pr)^2 - x^2 = (qr)^2 - \left( (pq)^2 - 2pqx + x^2 \right) . The x2x^2 terms cancel, so (pr)2(qr)2+(pq)2=2pqx (pr)^2 - (qr)^2 + (pq)^2 = 2pqx and x=(pr)2(qr)2+(pq)22pqx = \frac{(pr)^2 - (qr)^2 + (pq)^2}{2pq} .

We can now use the length of xx to find CFC'F', which is just (pr)2x2\sqrt{(pr)^2 - x^2}. The scale factor then is rCF\frac{r}{C'F'}, and since all lengths in the triangle are scaled by the same factor kk, the perimeter must also be scaled by kk.

Note by Toby M
3 months, 3 weeks ago

No vote yet
1 vote

  Easy Math Editor

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.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. list

  1. numbered
  2. 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"
MathAppears as
Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.
2 \times 3 2×3 2 \times 3
2^{34} 234 2^{34}
a_{i-1} ai1 a_{i-1}
\frac{2}{3} 23 \frac{2}{3}
\sqrt{2} 2 \sqrt{2}
\sum_{i=1}^3 i=13 \sum_{i=1}^3
\sin \theta sinθ \sin \theta
\boxed{123} 123 \boxed{123}

Comments

There are no comments in this discussion.

×

Problem Loading...

Note Loading...

Set Loading...