โ ๏ธA triangle has angles in the ratio 2:3:5. What are the measures of the angles?
๐To solve this question, we can set up an equation using the given ratio.
And, we know that the sum of the three angles of a triangle is 180ยฐ.
If we name the angles by the symbols: a, b and c. Then,
a + b + c = 180ยฐ
Given that
a/b=2/3 and b/c=3/5, we get
a= 2/3b and b= 3/5c, so,
a=2/3(3/5c)=2/5c.
Now, both a and b are expressed interms of c. So,
In the equation
a + b + c = 180ยฐ
Replace a and b to get,
2/5c + 3/5c + c = 180ยฐ
Which is one equation in one variable.
multiply both sides by 5 to get rid of the denominator :
2c + 3c + 5c = 5ร180
10c =900
c = 90
Then, b= 3/5c
=> b = 3/5(90)
= 3ร(90/5) = 3ร18 = 54,
and
a = 2/3b = 2/3(54)= 36.
โ Check: 36 + 54 + 90 = 180.
๐OR๐
Let x be the common multiplier for the ratio.
The measures of the angles can be represented as 2x, 3x, and 5x.
Since the sum of the angles in a triangle is 180 degrees, we can set up the equation:
2x + 3x + 5x = 180
Simplifying the equation:
10x = 180
Dividing both sides by 10:
x = 18
Now we can find the measures of the angles:
2x = 2 ร 18 = 36 degrees
3x = 3 ร 18 = 54 degrees
5x = 5 ร 18 = 90 degrees
Therefore, the measures of the angles in the triangle are 36 degrees, 54 degrees, and 90 degrees.
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