Altitudes Ratio [closed]
$begingroup$
If h, h', h'' denote the lengths of the three altitudes of a triangle, which of the following ratios never occurs as the ratio h: h': h''?
a)2 : 3 : 4
b)2 : 3 : 5
c)2 : 4 : 5
d)3 : 4 : 5
e)3 : 4 : 6
Any help would be much appreciated!
If possible, please could you explain the solution.
Thanks in Advance
triangle
asked Jul 16 '13 at 10:37
HummusHummus
1981312
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closed as off-topic by José Carlos Santos, Alexander Gruber♦ Dec 4 '18 at 4:07
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – José Carlos Santos, Alexander Gruber
If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
$begingroup$
If h, h', h'' denote the lengths of the three altitudes of a triangle, which of the following ratios never occurs as the ratio h: h': h''?
a)2 : 3 : 4
b)2 : 3 : 5
c)2 : 4 : 5
d)3 : 4 : 5
e)3 : 4 : 6
Any help would be much appreciated!
If possible, please could you explain the solution.
Thanks in Advance
triangle
asked Jul 16 '13 at 10:37
HummusHummus
1981312
$endgroup$
closed as off-topic by José Carlos Santos, Alexander Gruber♦ Dec 4 '18 at 4:07
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – José Carlos Santos, Alexander Gruber
If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
$begingroup$
If h, h', h'' denote the lengths of the three altitudes of a triangle, which of the following ratios never occurs as the ratio h: h': h''?
a)2 : 3 : 4
b)2 : 3 : 5
c)2 : 4 : 5
d)3 : 4 : 5
e)3 : 4 : 6
Any help would be much appreciated!
If possible, please could you explain the solution.
Thanks in Advance
triangle
asked Jul 16 '13 at 10:37
HummusHummus
1981312
$endgroup$
If h, h', h'' denote the lengths of the three altitudes of a triangle, which of the following ratios never occurs as the ratio h: h': h''?
a)2 : 3 : 4
b)2 : 3 : 5
c)2 : 4 : 5
d)3 : 4 : 5
e)3 : 4 : 6
Any help would be much appreciated!
If possible, please could you explain the solution.
Thanks in Advance
triangle
triangle
asked Jul 16 '13 at 10:37
HummusHummus
1981312
asked Jul 16 '13 at 10:37
HummusHummus
1981312
edited Jul 16 '13 at 10:56
Hummus
asked Jul 16 '13 at 10:37
HummusHummus
1981312
asked Jul 16 '13 at 10:37
HummusHummus
1981312
asked Jul 16 '13 at 10:37
HummusHummus
1981312
1981312
closed as off-topic by José Carlos Santos, Alexander Gruber♦ Dec 4 '18 at 4:07
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – José Carlos Santos, Alexander Gruber
If this question can be reworded to fit the rules in the help center, please edit the question.
closed as off-topic by José Carlos Santos, Alexander Gruber♦ Dec 4 '18 at 4:07
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – José Carlos Santos, Alexander Gruber
If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
Suppose that there is a triangle with altitudes $h_1$, $h_2$ and $h_3$.
Let $a_1$, $a_2$, and $a_3$ be the three sides of this triangle, such that the altitude from side $a_i$ is $h_i$. The area $A$ of the triangle is, by the well known formula:
$$
A=frac{1}{2}a_1h_1=frac{1}{2}a_2h_2=frac{1}{2}a_3h_3.
$$
With this, we use the condition that for a triangle to exist, the sum of the length of its shorter sides must be bigger than the length of its longer side. In terms of altitudes, if $h_1leq h_2leq h_3$, the condition thus becomes
$$
frac{1}{h_1} leq frac{1}{h_2}+frac{1}{h_3},
$$
which does not hold for b) only.
answered Jul 16 '13 at 11:40
zugggzuggg
1,074614
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add a comment |
$begingroup$
Sum of two altitudes is greater than the third altitude which is not satisfied by option (b).
As if we take them to be 2x,3x and 5x
2x+3x=5x(and it doesn't satisfy).
Cheers!
answered Dec 3 '18 at 17:50
maurana desanmaurana desan
1
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add a comment |
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Suppose that there is a triangle with altitudes $h_1$, $h_2$ and $h_3$.
Let $a_1$, $a_2$, and $a_3$ be the three sides of this triangle, such that the altitude from side $a_i$ is $h_i$. The area $A$ of the triangle is, by the well known formula:
$$
A=frac{1}{2}a_1h_1=frac{1}{2}a_2h_2=frac{1}{2}a_3h_3.
$$
With this, we use the condition that for a triangle to exist, the sum of the length of its shorter sides must be bigger than the length of its longer side. In terms of altitudes, if $h_1leq h_2leq h_3$, the condition thus becomes
$$
frac{1}{h_1} leq frac{1}{h_2}+frac{1}{h_3},
$$
which does not hold for b) only.
answered Jul 16 '13 at 11:40
zugggzuggg
1,074614
$endgroup$
add a comment |
$begingroup$
Suppose that there is a triangle with altitudes $h_1$, $h_2$ and $h_3$.
Let $a_1$, $a_2$, and $a_3$ be the three sides of this triangle, such that the altitude from side $a_i$ is $h_i$. The area $A$ of the triangle is, by the well known formula:
$$
A=frac{1}{2}a_1h_1=frac{1}{2}a_2h_2=frac{1}{2}a_3h_3.
$$
With this, we use the condition that for a triangle to exist, the sum of the length of its shorter sides must be bigger than the length of its longer side. In terms of altitudes, if $h_1leq h_2leq h_3$, the condition thus becomes
$$
frac{1}{h_1} leq frac{1}{h_2}+frac{1}{h_3},
$$
which does not hold for b) only.
answered Jul 16 '13 at 11:40
zugggzuggg
1,074614
$endgroup$
add a comment |
$begingroup$
Suppose that there is a triangle with altitudes $h_1$, $h_2$ and $h_3$.
Let $a_1$, $a_2$, and $a_3$ be the three sides of this triangle, such that the altitude from side $a_i$ is $h_i$. The area $A$ of the triangle is, by the well known formula:
$$
A=frac{1}{2}a_1h_1=frac{1}{2}a_2h_2=frac{1}{2}a_3h_3.
$$
With this, we use the condition that for a triangle to exist, the sum of the length of its shorter sides must be bigger than the length of its longer side. In terms of altitudes, if $h_1leq h_2leq h_3$, the condition thus becomes
$$
frac{1}{h_1} leq frac{1}{h_2}+frac{1}{h_3},
$$
which does not hold for b) only.
answered Jul 16 '13 at 11:40
zugggzuggg
1,074614
$endgroup$
Suppose that there is a triangle with altitudes $h_1$, $h_2$ and $h_3$.
Let $a_1$, $a_2$, and $a_3$ be the three sides of this triangle, such that the altitude from side $a_i$ is $h_i$. The area $A$ of the triangle is, by the well known formula:
$$
A=frac{1}{2}a_1h_1=frac{1}{2}a_2h_2=frac{1}{2}a_3h_3.
$$
With this, we use the condition that for a triangle to exist, the sum of the length of its shorter sides must be bigger than the length of its longer side. In terms of altitudes, if $h_1leq h_2leq h_3$, the condition thus becomes
$$
frac{1}{h_1} leq frac{1}{h_2}+frac{1}{h_3},
$$
which does not hold for b) only.
answered Jul 16 '13 at 11:40
zugggzuggg
1,074614
answered Jul 16 '13 at 11:40
zugggzuggg
1,074614
answered Jul 16 '13 at 11:40
zugggzuggg
1,074614
answered Jul 16 '13 at 11:40
zugggzuggg
1,074614
1,074614
add a comment |
add a comment |
$begingroup$
Sum of two altitudes is greater than the third altitude which is not satisfied by option (b).
As if we take them to be 2x,3x and 5x
2x+3x=5x(and it doesn't satisfy).
Cheers!
answered Dec 3 '18 at 17:50
maurana desanmaurana desan
1
$endgroup$
add a comment |
$begingroup$
Sum of two altitudes is greater than the third altitude which is not satisfied by option (b).
As if we take them to be 2x,3x and 5x
2x+3x=5x(and it doesn't satisfy).
Cheers!
answered Dec 3 '18 at 17:50
maurana desanmaurana desan
1
$endgroup$
add a comment |
$begingroup$
Sum of two altitudes is greater than the third altitude which is not satisfied by option (b).
As if we take them to be 2x,3x and 5x
2x+3x=5x(and it doesn't satisfy).
Cheers!
answered Dec 3 '18 at 17:50
maurana desanmaurana desan
1
$endgroup$
Sum of two altitudes is greater than the third altitude which is not satisfied by option (b).
As if we take them to be 2x,3x and 5x
2x+3x=5x(and it doesn't satisfy).
Cheers!
answered Dec 3 '18 at 17:50
maurana desanmaurana desan
1
answered Dec 3 '18 at 17:50
maurana desanmaurana desan
1
answered Dec 3 '18 at 17:50
maurana desanmaurana desan
1
answered Dec 3 '18 at 17:50
maurana desanmaurana desan
1
1
add a comment |
add a comment |
