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
Short-Answer Question:
If $\alpha$ and $\beta$ are the zeroes of a polynomial $f(x) = x^2 + x - 2$, find the value of $\Big(\frac{1}{\alpha}-\frac{1}{\beta}\Big).$
→If $\alpha$ and $\beta$ are the zeroes of a polynomial $f(x) = x^2 + x - 2$, find the value of $\Big(\frac{1}{\alpha}-\frac{1}{\beta}\Big).$
In the given figure, AB is a chord of length 16cm of a circle of radius 10cm. The tangents at A and B intersect at a point P. Find the length of PA.
The midpoint of the line segment joining A(2a, 4) and B(-2, 3b) is C(1, 2a + 1). Find the values of a and b.
A 5-m-wide cloth is used to make a conical tent of base diameter 14m and height 24m. Find the cost of cloth used, at the rate of ₹ 25 per metre.
Find the probability of getting the sum of two numbers, less than 3 or more than 11, when a pair of distinct dice is thrown together.
The points $A(x_1, y_1), B(x_2, y_2)$ and $C(x_3, y_3)$ are the vertices of $\triangle\text{ABC}.$
Find the coordinates of points Q and R on medians BE and CF, respectively such that BQ : QE = 2 : 1 and CR : RF = 2 : 1.
→Find the coordinates of points Q and R on medians BE and CF, respectively such that BQ : QE = 2 : 1 and CR : RF = 2 : 1.
A sailor goes 8km downstream in 40 minutes and returns in 1 hour. Find the speed of the sailor in still water and the speed of the current.
Find the $12^{th}$ term from the end of the following arithmetic progressions:
$1, 4, 7, 10, ....., 88.$
→$1, 4, 7, 10, ....., 88.$
Find the common difference of the A.P. and write the next two terms:
$0,\frac{1}{4},\frac{1}{2},\frac{3}{4}, .....$
→$0,\frac{1}{4},\frac{1}{2},\frac{3}{4}, .....$
If G be the centroid of a triangle ABC and P be any other point in the plane, prove that $PA^2 + PB^2 + PC^2 = GA^2 + GB^2 + GC^2 + 3GP^2.$
→