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1. The circle (x + 3)2 + (y − 4)2 = 36 and the line 3x − 4y + 10 = 0 intersect twice as shown below. What is the distance from the center of the circle to the line?

2. A line goes through (2, 4) and (−8,−1). The line has its x-intercept at point P and its y-intercept at point Q. Find the area of △OPQ, where O is the origin.

3.In △ABC, AB = 4 and BC = 5. Find the number of possible integer values of CA. Reminder: In this class, we do not consider degenerate triangles (i.e. a triangle whose area is 0).

4.In obtuse △ABC, AB = 4 and BC = 5. Find the number of possible integer values of CA.

5. A, B, C, and D are points on a circle, and segments AC and BD intersect at a point P inside the circle, such that AP = 8, PC = 9, and BD = 22. Find BP, given that BP < DP.

6.] In △ABC, sin A : sin B : sinC = 4 : 5 : 6. Find cos(A + C).

THank youuu

 Nov 5, 2023
 #2
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Problem 2

 

First we will find the slopes of the lines that go through P and Q respectively. The slope of the first line is xP​−xO​yP​−yO​​=4. The slope of the second line is xQ​−xO​yQ​−yO​​=−1. We know the angle between the lines is 180 degrees because they are perpendicular to each other. Then, the slope of the first line is −1 times the reciprocal of the slope of the second line which means the slope is −11​ which is −1. Now we can find the equations of the lines. The equation of the line that goes through P is y−4=−1(x−2) and the equation of the line that goes through Q is y−0=−1(x+8). We can find the coordinates of P by letting y=0 in the first equation. We get 0−4=−1(x−2) and x=6. We can find the coordinates of Q by letting x=0 in the second equation. We get y−0=−1(0+8) and y=−8. The area of triangle OPQ=8*6/2​=24​.

 Nov 5, 2023
 #3
avatar+1612 
+1

Problem 3

 

By the triangle inequality, (5-4 \leq CA \leq 5+4), so 1≤CA≤9. Therefore, there are 9​ possible integer values of CA.

 

Problem 4

 

 

By the Triangle Inequality, we know that 3 <9. We also know that the area of an obtuse triangle is positive, so CA=4 and CA=5. Therefore, the only possible integer values of CA are 3,6,7,8​.

 Nov 5, 2023

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