With an extended ratio you can express the lengths of the sides with respect to the factor x. You will need one of the trigonometric ratios to find the measure of the angle.
B
Practice makes perfect
An extended ratio compares three or more numbers. In an extended ratio a : b : c, the ratio of the first two numbers is a : b, the ratio of the last two numbers is b : c, and the ratio of the first and last numbers is a : c. We want to find the measure of the smallest angle of a triangle, whose sides fit the given extended ratio.
5 : 12 : 13
This means that we can express the lengths of the sides of the triangle as 5x, 12x, and 13x.
Knowing that, we can determine whether our triangle is acute, right, or obtuse. To do so, we will compare the square of the largest side length to the sum of the squares of the other two side lengths. Let a, b, and c be the lengths of the sides, with c being the longest.
Condition
Type of Triangle
a^2+b^2 < c^2
Obtuse triangle
a^2+b^2 = c^2
Right triangle
a^2+b^2 > c^2
Acute triangle
Let's now consider the side lengths 5x, 12x, and 13x. Since 13x is the greatest, we will let c be 13x. We will also arbitrarily let a be 5x and b be 12x.
( 5x)^2+(12x)^2 ? ( 13x)^2
Let's simplify the above statement to determine whether the left-hand side is less than, equal to, or greater than the right-hand side.
Referring back to our table, we can conclude that our triangle with side lengths 5x, 12x, and 13x is a right triangle. In this type of triangle, one angle is a right angle and the other two are acute. The right angle and the hypotenuse are opposite each other. In our case, the hypotenuse has the length of 13x, since it is the greatest side length.
The smallest angle is opposite the shortest side of the triangle. Therefore, we want to find the measure of the angle that is opposite the leg with side length of 5x. Let's call this measure y.
Now, let's use one of the trigonometric ratios to find the value of y. We will use the expressions for the lengths of the opposite and the adjacent sides to the unknown angle. To find its measure we can use the tangent ratio.
tan y = Length of leg opposite toy/Length of leg adjacent toy
In our triangle, we have that the length of the opposite and adjacent legs to y are 5x and 12x.
The tangent of the angle is 512. Now, to isolate y we will use the inverse function of tan.
tan y=5/12 ⇔ y=tan ^(- 1)5/12
Let's use a calculator to find the value of tan ^(- 1) 512. First, we will set our calculator into degree mode. To do so, push MODE, select Degree instead of Radian in the third row, and push ENTER. Next, we push 2ND followed by TAN, introduce the value 512, and press ENTER.
The angle is about 22.62, which corresponds to option B.
