A secant and a tangent to a circle intersect in a 42 degree angle. The two arcs of the circle intercepted by the secant and tangent have measures in a 7:3 ratio. Find the measure of the third arc.

If someone could help me figure out how to do the equations, then that would be great. Thank you.

42° = 1/2 (large-small)

so, large-small=84°
but, large = 7/3 * small
7/3 * small = small+84
4/3 * small = 84
small = 63
large = 147

So, since the three arcs add up to 360°,

63+147+x = 360
x = 150°

A nice discussion of the topic is found at

http://www.mathwarehouse.com/geometry/circle/tangents-secants-arcs-angles.php

given ab is a tangent of point x, ab=70 and xd= 12 what is the measure of db?

amosc @ unique4418

Well, this problem seems like a real angle grinder! Let's see if we can find the measure of the third arc together.

First, let's call the measure of the larger arc intercepted by the secant and tangent as 7x, and the measure of the smaller arc as 3x.

Since the secant and tangent intersect in a 42-degree angle, we can use one of the pretzel-like properties of a circle to find the measure of each of the intercepted arcs.

The angle formed by the secant and its intercepted arc is half the measure of the intercepted arc. So, the angle formed by the smaller arc is 3x/2 degrees. Similarly, the angle formed by the larger arc is 7x/2 degrees.

Now it's time to unleash some math skills! Since the secant and tangent intersect in a 42-degree angle, we know that the angles formed by the secant (3x/2) and tangent (7x/2) must add up to 42.

So, we can write an equation: (3x/2) + (7x/2) = 42.

To simplify, we can combine like terms: (10x/2) = 42.

Dividing both sides by 5, we find that x = 8.4.

Now we can determine the measure of the third arc. Since the three arcs add up to 360 degrees in a circle, we have:

3x + 7x + Third Arc = 360.

Simplifying, we get 10x + Third Arc = 360.

Substituting the value of x we found earlier, we have 10 * 8.4 + Third Arc = 360.

So, 84 + Third Arc = 360.

Finally, we can subtract 84 from both sides to find the measure of the third arc:

Third Arc = 360 - 84.

Calculating that, we have the answer: The measure of the third arc is 276 degrees.

I hope this explanation brought some joy to your mathematical journey!

To solve this problem, there are a few steps we can follow:

Step 1: Understand the problem
We are given that a secant and a tangent intersect in a 42-degree angle. In addition, the measures of the two arcs intercepted by the secant and tangent have a 7:3 ratio.

Step 2: Determine the variables
Let's assign variables to the unknowns in the problem. Let's call the smaller arc measure as 7x, the larger arc measure as 3x, and the third arc measure (the one we need to find) as y.

Step 3: Formulate equations
Using the information given, we can form two equations:

Equation 1: The sum of the angles formed by the secant and tangent is 180 degrees. Since we are given that the angle formed is 42 degrees, the equation becomes:
7x + 3x = 42

Equation 2: The sum of the measures of the intercepted arcs is equal to the measure of the third arc. So, we have:
7x + 3x = y

Step 4: Solve the equations
We can solve the equations simultaneously:

From Equation 1:
7x + 3x = 42
10x = 42
x = 42/10
x = 4.2

Substitute the value of x into Equation 2 to find y:
7x + 3x = y
7(4.2) + 3(4.2) = y
29.4 + 12.6 = y
42 = y

So, the measure of the third arc is 42 degrees.

Step 5: Check the solution
We can verify our solution by ensuring that the ratio of the intercepted arcs satisfies the given 7:3 ratio:
7x = 7(4.2) = 29.4
3x = 3(4.2) = 12.6
The ratio of 29.4 to 12.6 simplifies to 7:3, which matches the given ratio.

Therefore, the measure of the third arc is 42 degrees.