c Connect the left ends of chords together with a segment. Also, connect the right ends of chords with a segment. Then, find a pair of similar triangles.
A
a 270^(∘)
B
b x = 132^(∘), y ≈ 15.7
C
c x=2
Practice makes perfect
a We want to find the value of x from the following diagram.
Here, x is the measure of the longer arc that is formed from between the two points of tangency of the circle and the two line segments. Let's call the center of our circle C and add two line segments that connect C to the points of tangency. The angles formed by the added line segments and the existing ones are right angles.
We have a quadrilateral with 3 right angles and a missing angle near C. The sum of interior angles of a quadrilateral equals 360^(∘). Let's write an equation for m ∠ C using this fact and solve it.
The angle with x degrees and ∠ C are adjacent angles that form a complete angle. Since we know that m ∠ C = 90^(∘), we can use this fact to write an equation for x and solve it.
x + 90^(∘) = 360^(∘) ⟹ x = 270^(∘)
b We want to find the values of x and y from the following diagram.
Point C is the center of the circle, so the dashed line segments are the radii of that circle. In consequence, these segments are congruent. The radii intersect the rays at the points of tangency of rays and the circle, so the angles they make with the rays are right angles. Let's add this information to our diagram.
The sum of the interior angles of a quadrilateral equals 360^(∘). We can use this fact to write an equation for x.
48^(∘) + x + 90^(∘) + 90^(∘) = 360^(∘)
Let's solve it!
We found that x = 132^(∘). Now, we will find the value of y. Let's connect the vertex of the 48^(∘) angle and point C with a line segment. This segment bisects the 48^(∘) angle into two 48^(∘) ÷ 2 = 24^(∘) angles.
We see that y is a side length of a leg adjacent to the 24^(∘) angle. The leg opposite that angle is 7 units long. Let's recall the definition of a cotangent ratio.
cot θ = Adjacent/Opposite
Let's substitute 24^(∘) for θ, y for the length of the adjacent leg, and 7 for the length of the opposite leg. This will give us an equation that we can solve for y.
c We want to find the value of x using the following diagram.
To do so, let's connect the left and the right endpoints of both chords with a segment. Also, let's label the endpoints and the intersection of the original two chords.
Now, notice that ∠ DAC and ∠ DBC are two inscribed angles which intercept the same arc DC. As a result, these angles are congruent. Similarly, ∠ ADB and ∠ ACB intercept the same arc AB, so the two angles are congruent. We conclude that △ ADE is similar to △ BCE by the Angle-Angle Similarity Theorem.
△ AED ~ △ BEC
Since the triangles are similar, the ratios of the corresponding sides are equal. In our case, BE corresponds to AE and EC corresponds to ED.
BE/AE = EC/ED
Let's substitute the lengths from our diagram into the above expression to write an equation for x.
x/3 = x+2/6
Let's solve this equation!
