Maths-
General
Easy
Question
If the angles of a triangle are in ratio 4 : 1 : 1 then the ratio of the longest side and perimeter of triangle is -
- None of these
The correct answer is: ![fraction numerator square root of 3 over denominator 2 plus square root of 3 end fraction](data:image/png;base64,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)
Given that the angles of a triangle are in ratio 4 : 1 : 1 then we have to find the ratio of the longest side and perimeter of triangle.
Given: Ratio of angles of a triangle =4:1:1
Let the angles of a triangle are 4x,x and x respectively
∴4x+x+x=180∘
⇒x=30
∠A=4x=4×30=120
∠B=x=30
and ∠C=x=30
By sine Law, we have
a/sinA=b/sinB=c/sinC=k (assume)
a=ksinA=k×sin120∘=k×sin(90+30)=kcos30=(√3/2)k
b=ksinB=ksin30=k×1/2=k/2
c=ksinC=ksin30∘=k×1/2=k/2
∠A is the biggest angle.
So, the largest side will be BC. [Side opposite to biggest angle is longest.]
Ratio of longest side to the perimeter =a/(a+b+b)
![fraction numerator begin display style fraction numerator square root of 3 k over denominator 2 end fraction end style over denominator fraction numerator square root of 3 k over denominator 2 end fraction plus begin display style k over 2 end style plus k over 2 end fraction
fraction numerator fraction numerator square root of 3 over denominator 2 end fraction over denominator fraction numerator square root of 3 over denominator 2 end fraction plus begin display style 2 over 2 end style end fraction
fraction numerator square root of 3 over denominator square root of 3 plus 2 end fraction](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAG8AAADkCAYAAACWsnp7AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAABwkCHjogAAByRJREFUeNrtnVGoFFUYxw8iIiGRlIQkKCIRIeKDopIiF0JEfBC5glIhloT05Fs+CPkSvgRK9hIaEoFBZEVcLIiQkJArkRE9iELSkw+CgkFeArXvY79L67AzO7t398z3Ob8f/HHdnd27/M/OmXNmvvmflOKzSPRoDoIG2Su6iA0x+U50EBvi8Zzorv0LwTgk+rrH8xOiC6J/RDOi66KTomf7fJ5uPw9b86DHusmS53eLFnQ9t1b0fcVnrRL9jqV5WCG6LVo4wHvuVby2S/Q5tubhiOj0ANuvFF2reP090bv2eL7oU9EObB4PV0Xba2z3vOiAHfeqGuML2/sW2uMJLB7NJLzIatEt20PKKE7GD/f5O9ftc78VbcD2uaEDiLNm/ObCa8dFJ2p+zmLbo6ZFW0u2mWcjzcvWgDBHjlrj6TzuSuG1m6J1Q+zB0yWvbbAR6j7RJ1g/Ot4RPbRjl7JRdKNPl1nGTMnz++yHonxg80cYEdqlnbPHp0THhviMNXZc68WHore7/v+D6BVsHw1nRPdtb9O53Ut9tv9JtMe2n2cjR+1q95ds/1VhJLrYutilWD93nhE9EJ23QUU/toimrJucsePZzort7/SY7OvA5ccBTwJACZdqDvnBIeut8ZZhRUwmsQAAoMgj5wIAAACANqNXA75MnWKjf0W/iV7DlhjoFQW9PjdbPvGy6Gd7DgKyPFGTGZoZLIjJJus6IRh6MXU6UdYQDi1n+Ea0DStisdIabhVWxEILlPQ+hqewIhZa36n3GszHinhMpf6lgeAULqQCAAAARBulIcoPAQAAAGBQWp3yp/eKXwo6xG19yt/rqZM0G5HWp/xpUGnUiOBWp/xpl/l3ejwieLZ7nC0z31bDuKbwlPKX3Y9+XaZezf6rj3FN4inlL7sf/bpMHcXdqTDuBXusBbFaurAk43f3lvI3Vj8W1egyu9Eg7o9SJ6S0zDjlLfvlL8pslqeUv7H68aodx9bU7DJnj3tvVBin04uPRZ81NM/ylPI3Vj+O2HzozwG6zKdF76fHk/S6jdP7535J5UFu48ZTyl8WP3TXPlyjy+xmpsK4JdbXb2rAME8pf1n8OGgN+GbNifkW+zVVGacl6X+k/4NQc+Ep5S+bH9fseHawz/FO9zhdgGJ5DeM22eChrdGI2fx40RrHy9o9ns6XhihP2I5hcRsvYRiNR+ON+ct4K20b93u9fhf2PLpNGo/Go/GAxhvDIEevpGtoDUkMAT3Tyxq6gNOvTg3zmAzozjOveV+ekwFdeLbVTIqCh2RAF54ttV/yymDHn5m2e6ZXHqZSvOXLmkwGdOGZ/vELqVMGEYkmkwHdeKZfIlr6UNPJgG48i3ZS1UMyIPecDwHJgEEhGTAwJAMGhowTAAAAAAAAAABwROQMs9YTOcOs9UTOMGt9lzlshhk47zKrMszAeZdZlWEGGRllhhlkZNQZZpCRUWeYQQOMKsMMGmBUGWbQEKPIMIOG8JZhBgOyHQsAAKoGHyy/CQAAAE/EoIbMssB4zyyDGnAVISjRMsvAiJpZ1nqiZpaxx6WYmWWQYmaWQWGex0lnAAAAGHjQ0EYBAAAAAFTjcZUud3gNv/G8SpcbIoXfeFilyxXRwm+ojenqMiOF3zS5Sle4LtNT+E2Tq3SF7DK9hN80vUpX40QNv/GwSlejRA2/YZWuFDP8hlW6CkQKv2GVrgKRwm+42NoDwm8Ckzv8hrKEEZMz/IbGCwyNR+O107jc5XXjfi+jT/Y8oPFoPKDxWjyoIfwmMN7Cbyj9GwIvBT6U/g2I9/AbSv9KiBJ+Q+lfgSjhN5T+9djjIoTfUPrXgwjhN60v/es3z/N6orf1pX9RofQvKJT+BYbSvyfsWMzFVwAAAAAAAAAAAICWouV4BHgHZa/oIjbERJMkDmJDPDSI4G7KF0gAI+RQ6qQnFZlInUo0zXnRwtjropOpk2EGTtBj3WTJ87tFC7qeW5s6WS7ggBWi26lTAFuXe9jmAw2gOz3A9lpzeQ3bfHA11Qvg0bK9A3bc24Ft+SfhRVaLbqXqOsriZPwwVuZDS8rPmvGbC68dF52o+Tl6b8Gu1LkxZCu25uGoNZ7O464UXrspWjfEHjyNrXnRe9Mf2rFL2Si6kYYrPedmyAbQyfY5e3xKdGyIz1hjgxbIzBnRfdvbdG7X774BvZd9j22vyRIT1tXux8r8aA71A9F50eUa22uk8ZR1kyo947ITG5vjEkP+uKy3xluGFTGZxAIAeHJ51AIBAAAAQBQoWw8MZeuBoWw9KJStB8ZT2Tox/APiqWydGP4BWJHylK3PZURKDH8JucrW5zqdoPKsB7nK1ufSeK2P4W+6bH3Yxmt1DH9TZeujOCvT+hh+L2Xrg+55xPB30XTZ+iCNRwx/D5osW6/beMTwl9Bk2XrdxiOGv4QIZetcdqqAsvXAULYeHMrWHfIfcXkfz0u0VLcAAAL/dEVYdE1hdGhNTAA8bWF0aCB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMTk5OC9NYXRoL01hdGhNTCI+PG1mcmFjPjxtc3R5bGUgZGlzcGxheXN0eWxlPSJ0cnVlIj48bWZyYWM+PG1yb3c+PG1zcXJ0Pjxtbj4zPC9tbj48L21zcXJ0PjxtaT5rPC9taT48L21yb3c+PG1uPjI8L21uPjwvbWZyYWM+PC9tc3R5bGU+PG1yb3c+PG1mcmFjPjxtcm93Pjxtc3FydD48bW4+MzwvbW4+PC9tc3FydD48bWk+azwvbWk+PC9tcm93Pjxtbj4yPC9tbj48L21mcmFjPjxtbz4rPC9tbz48bXN0eWxlIGRpc3BsYXlzdHlsZT0idHJ1ZSI+PG1mcmFjPjxtaT5rPC9taT48bW4+MjwvbW4+PC9tZnJhYz48L21zdHlsZT48bW8+KzwvbW8+PG1mcmFjPjxtaT5rPC9taT48bW4+MjwvbW4+PC9tZnJhYz48L21yb3c+PC9tZnJhYz48bXNwYWNlIGxpbmVicmVhaz0ibmV3bGluZSIvPjxtZnJhYz48bWZyYWM+PG1zcXJ0Pjxtbj4zPC9tbj48L21zcXJ0Pjxtbj4yPC9tbj48L21mcmFjPjxtcm93PjxtZnJhYz48bXNxcnQ+PG1uPjM8L21uPjwvbXNxcnQ+PG1uPjI8L21uPjwvbWZyYWM+PG1vPis8L21vPjxtc3R5bGUgZGlzcGxheXN0eWxlPSJ0cnVlIj48bWZyYWM+PG1uPjI8L21uPjxtbj4yPC9tbj48L21mcmFjPjwvbXN0eWxlPjwvbXJvdz48L21mcmFjPjxtc3BhY2UgbGluZWJyZWFrPSJuZXdsaW5lIi8+PG1mcmFjPjxtc3FydD48bW4+MzwvbW4+PC9tc3FydD48bXJvdz48bXNxcnQ+PG1uPjM8L21uPjwvbXNxcnQ+PG1vPis8L21vPjxtbj4yPC9tbj48L21yb3c+PC9tZnJhYz48L21hdGg+cQLNgQAAAABJRU5ErkJggg==)
So, the ratio of the longest side and perimeter of triangle is
Related Questions to study
maths-
In a triangle PQR as shown in figure given that x : y : z :: 2 : 3 : 6, then the value of
QPR is -
![](data:image/png;base64,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)
In a triangle PQR as shown in figure given that x : y : z :: 2 : 3 : 6, then the value of
QPR is -
![](data:image/png;base64,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)
maths-General
maths-
If the sides be a, b, c then (r + r1) tan
+ (r + r2) tan
+ (r + r3) tan
=
If the sides be a, b, c then (r + r1) tan
+ (r + r2) tan
+ (r + r3) tan
=
maths-General
Maths-
If in a triangle ABC; AD, BE and CF are the altitudes and R is the circum-radius, then the radius of the circle DEF is -
Therefore, the radius of the circle DEF is R/2.
If in a triangle ABC; AD, BE and CF are the altitudes and R is the circum-radius, then the radius of the circle DEF is -
Maths-General
Therefore, the radius of the circle DEF is R/2.
Maths-
In an equilateral triangle of side
cms, the circum radius is -
The circumradius is 2cm.
In an equilateral triangle of side
cms, the circum radius is -
Maths-General
The circumradius is 2cm.
maths-
If the lines
and
are concurrent then
If the lines
and
are concurrent then
maths-General
maths-
is a unit vector such that
0 then a unit vector
perpendicular to ![text both end text stack a with rightwards arrow on top text and end text stack c with rightwards arrow on top text end text text is end text](data:image/png;base64,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)
is a unit vector such that
0 then a unit vector
perpendicular to ![text both end text stack a with rightwards arrow on top text and end text stack c with rightwards arrow on top text end text text is end text](data:image/png;base64,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)
maths-General
maths-
The reflection of the point (2, –1, 3) in the plane 3x – 2y – z = 9 is :
The reflection of the point (2, –1, 3) in the plane 3x – 2y – z = 9 is :
maths-General
maths-
A, B, C & D are four points in a plane with position vectors
respectively such that
Then for the triangle ABC, D is its:
A, B, C & D are four points in a plane with position vectors
respectively such that
Then for the triangle ABC, D is its:
maths-General
maths-
Equation of the angle bisector of the angle between the lines
![fraction numerator x minus 1 over denominator 1 end fraction equals fraction numerator y minus 2 over denominator 1 end fraction equals fraction numerator z minus 3 over denominator negative 1 end fraction text is : end text](data:image/png;base64,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)
Equation of the angle bisector of the angle between the lines
![fraction numerator x minus 1 over denominator 1 end fraction equals fraction numerator y minus 2 over denominator 1 end fraction equals fraction numerator z minus 3 over denominator negative 1 end fraction text is : end text](data:image/png;base64,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)
maths-General
maths-
The sine of angle formed by the lateral face ADC and plane of the base ABC of the tetrahedron ABCD where A º (3, –2, 1) ; B º (3, 1, 5); C º (4, 0, 3) and D º (1, 0, 0) is -
The sine of angle formed by the lateral face ADC and plane of the base ABC of the tetrahedron ABCD where A º (3, –2, 1) ; B º (3, 1, 5); C º (4, 0, 3) and D º (1, 0, 0) is -
maths-General
maths-
Four coplanar forces are applied at a point O. Each of them is equal to k and the angle between two consecutive forces equals 45º as shown in the figure. Then the resultant has the magnitude equal to :
![](data:image/png;base64,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)
Four coplanar forces are applied at a point O. Each of them is equal to k and the angle between two consecutive forces equals 45º as shown in the figure. Then the resultant has the magnitude equal to :
![](data:image/png;base64,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)
maths-General
maths-
be vectors of length 3,4,5 respectively.
be perpendicular to
and
is equal to :
be vectors of length 3,4,5 respectively.
be perpendicular to
and
is equal to :
maths-General
maths-
and
is equal to :
and
is equal to :
maths-General
maths-
If ABCDEF is a regular hexagon and if
then ![text end text lambda text is - end text](data:image/png;base64,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)
If ABCDEF is a regular hexagon and if
then ![text end text lambda text is - end text](data:image/png;base64,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)
maths-General
maths-
If
is along the angle bisector of
then -
If
is along the angle bisector of
then -
maths-General