Given a right-angled triangle with perimeter $18$. The sum of the squares of the three side lengths is $128$. What is the area of the triangle?
Problem
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Tags: area of a triangle, right triangles
ilovepizza2020
19.09.2020 02:52
We have that $c^2 = 64$ so $c = 8$. Then, we have $a+b = 10$, $a^2+b^2 = 64$, so $\frac{ab}{2} = \frac{1}{4}\cdot(10^2-64) = 9$
franzliszt
19.09.2020 03:01
$a+b+c=18$ and $a^2+b^2+c^2=128$ and $a^2+b^2=c^2$. Obviously, $c=8$. Substitute to get $a+b=10 \iff a^2+2ab+b^2=100$ and $a^2+b^2=64$. Hence, $2ab=36 \iff ab=18$. So the answer is $9$.
Aspiring_Mathletes
19.09.2020 03:05
Isn't this too basic for high school math?
ilovepizza2020
19.09.2020 03:06
Aspiring_Mathletes wrote: Isn't this too basic for high school math? Probably? But you can just report OP if you think that.
franzliszt
19.09.2020 03:08
Well maybe it was a high school contest in Malaysia. Still some nice problems from this contest!
ilovepizza2020
19.09.2020 03:09
franzliszt wrote: Well maybe it was a high school contest in Malaysia. Still some nice problems from this contest! And nice way to postfarm
franzliszt
19.09.2020 03:11
Well I don't really care about post farming at this point given how many I already have lol
Keith50
19.09.2020 03:14
franzliszt wrote: Well maybe it was a high school contest in Malaysia. Still some nice problems from this contest! Actually this was from Malaysia IMO National Selection Test 1:) but it's easy.
ilovepizza2020
19.09.2020 03:16
Keith50 wrote: franzliszt wrote: Well maybe it was a high school contest in Malaysia. Still some nice problems from this contest! Actually this was from Malaysia IMO National Selection Test 1:) but it's easy. Oh wow this is like a TST! Wow!
franzliszt
19.09.2020 03:17
Maybe first round like equiv to AMC 12?
Keith50
19.09.2020 03:17
ilovepizza2020 wrote: Keith50 wrote: franzliszt wrote: Well maybe it was a high school contest in Malaysia. Still some nice problems from this contest! Actually this was from Malaysia IMO National Selection Test 1:) but it's easy. Oh wow this is like a TST! Wow! Nah, it's like AMC but a lot easier.
Aspiring_Mathletes
19.09.2020 03:34
Keith50 wrote: franzliszt wrote: Well maybe it was a high school contest in Malaysia. Still some nice problems from this contest! Actually this was from Malaysia IMO National Selection Test 1:) but it's easy. Nice! What are the rounds in the IMO Selection Process? Sorry if this is off-topic, I'm just curious
riemann-1
19.09.2020 04:15
Keith50 wrote: Given a right-angled triangle with perimeter $18$. The sum of the squares of the three side lengths is $128$. What is the area of the triangle? Too easy
hopeless404
19.09.2020 04:27
Despite the name, IMONST is just the first part of Malaysia's selection process, designed so that areas with lesser education still get to participate (and do well-ish?)