Question
In a $\triangle\text{ABC},$ $\angle\text{x}^\circ,\angle\text{B}=(3\text{x}-2)^\circ,\angle\text{C}=\text{y}^\circ$ Also $\angle\text{C}-\angle\text{B}=9^\circ.$ Find the three angles.
✓
Answer
It is given that,
$\angle\text{A}=\text{x}^\circ ....(\text{i})$
$\angle\text{B}=(3\text{x}-2)^\circ\ ...(\text{ii})$
$\angle\text{C}=\text{y}^\circ\ ....(\text{iii})$
And, $\angle\text{C}-\angle\text{B}=9^\circ\ ....(\text{iv})$
Putting $\angle\text{C}=\text{y}^ \circ$ and $\angle\text{B}=(3\text{x}-2)^\circ$ in equation (iv), we get
y - (3x - 2) = 9
⇒ y - 3x + 2 = 9
⇒ y - 3x = 9 - 2
⇒ -3x + y = 7 .....(v)
We know that, the sum of angles of a triangle is 180°
$\therefore\angle\text{A}+\angle\text{B}+\angle\text{C}=180^\circ$
⇒ x + 3x - 2 + y = 180°
⇒ 4x - 2 + y = 180°
⇒ 4x + y = 180 + 2
⇒ 4x + y = 182 .......(vi)
Subtracting equation (v) from equatoin (vi) we get
4x + 3x = 182 - 7
⇒ 7x = 175
$\Rightarrow\text{x}=\frac{175}{7}$
⇒ x = 25
Putting x = 25 in equation (v) we get
-3 × 25 + y = 7
⇒ -75 + y = 7
⇒ y = 7 + 75
⇒ y = 82
$\therefore\angle\text{A}=\text{x}^\circ=25^\circ$
$\angle\text{B}=(3\text{x}-2)^\circ$
(3 × 25 - 2)°
= (75 - 2)
= 73°
And, $\angle\text{C}=\text{y}^\circ=82^\circ$
$\angle\text{A}=\text{x}^\circ ....(\text{i})$
$\angle\text{B}=(3\text{x}-2)^\circ\ ...(\text{ii})$
$\angle\text{C}=\text{y}^\circ\ ....(\text{iii})$
And, $\angle\text{C}-\angle\text{B}=9^\circ\ ....(\text{iv})$
Putting $\angle\text{C}=\text{y}^ \circ$ and $\angle\text{B}=(3\text{x}-2)^\circ$ in equation (iv), we get
y - (3x - 2) = 9
⇒ y - 3x + 2 = 9
⇒ y - 3x = 9 - 2
⇒ -3x + y = 7 .....(v)
We know that, the sum of angles of a triangle is 180°
$\therefore\angle\text{A}+\angle\text{B}+\angle\text{C}=180^\circ$
⇒ x + 3x - 2 + y = 180°
⇒ 4x - 2 + y = 180°
⇒ 4x + y = 180 + 2
⇒ 4x + y = 182 .......(vi)
Subtracting equation (v) from equatoin (vi) we get
4x + 3x = 182 - 7
⇒ 7x = 175
$\Rightarrow\text{x}=\frac{175}{7}$
⇒ x = 25
Putting x = 25 in equation (v) we get
-3 × 25 + y = 7
⇒ -75 + y = 7
⇒ y = 7 + 75
⇒ y = 82
$\therefore\angle\text{A}=\text{x}^\circ=25^\circ$
$\angle\text{B}=(3\text{x}-2)^\circ$
(3 × 25 - 2)°
= (75 - 2)
= 73°
And, $\angle\text{C}=\text{y}^\circ=82^\circ$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
In the following figure, ABCD is a square of side 2a, Find the ratio between The circumferences. The areas of the in circle and the circum-circle of the square.
In the adjoining figure, ABC is a triangle in which AB = AC. If D and E are points on AB and AC respectively such that AD = AE, show that the points B, C, E and D are concyclic.
Solve the following system of equations by the method of cross-multiplication:
$\frac{\text{x}}{\text{b}}=\frac{\text{y}}{\text{b}},$
$\text{ax}+\text{by}=\text{a}^2+\text{b}^2.$
→$\frac{\text{x}}{\text{b}}=\frac{\text{y}}{\text{b}},$
$\text{ax}+\text{by}=\text{a}^2+\text{b}^2.$
In the following, determine whether the given values are solution of the given equation or not:
$\text{x}+\frac{1}{\text{x}}=\frac{13}{6},$ $\text{x}=\frac{5}{6},\text{x}=\frac{4}{3}$
→$\text{x}+\frac{1}{\text{x}}=\frac{13}{6},$ $\text{x}=\frac{5}{6},\text{x}=\frac{4}{3}$
Solve : $\frac{4}{x}+\frac{3}{y}=1 ; \frac{8}{x}-\frac{9}{y}=7$
→Evaluate the following:
In the adjoining figure, $\triangle\text{ABC}$ is a right-angled at B and $\angle\text{A}=45^\circ,$ If $\text{AC}=3\sqrt{2}\text{cm},$
Find:
→In the adjoining figure, $\triangle\text{ABC}$ is a right-angled at B and $\angle\text{A}=45^\circ,$ If $\text{AC}=3\sqrt{2}\text{cm},$
Find:
- BC.
- AB.
A hemispherical bowl of internal radius 9cm is full of water. Its contents are emptied into a cylindrical vessel of internal radius 6cm. Find the height of water in the cylindrical vessel.
If the radius of a sphere is doubled, what will be the ratio of its surface area and volume as to that of the first?
Find the area of a parallelogram ABCD if three of its vertices are A(2, 4), $\text{B}(2+\sqrt{3},\ 5)$ and C(2, 6).
→The rain water from a roof of 44m × 20m drains into a cylindrical tank having diameter of base 4m and height 3.5m. If the tank is just full, then find the rainfall in cm.