1. the points given are two of the 3 vertices of a right triangle. Which could be the third vertex of the triangle given 2 points are (-4,-5) & (3,2) ?
2.) A circle with center O and radius r is intersected at 2 distinct points by line l. Which is true?
a. The distance between the points of intersection is less than r
b.There is at least 1 point on l whose distance from O is less than r
c. There is no point on l whose distance from O is less than 1/2
3.) In triangle ABC, the measure of A is = the measure of C. If the length of AB is x, the length of AC is x-2, and the length of BC is 3x-8, what is the length of AC?
Answer? 5
#1
If you make a sketch and assume the two points form the hypotenuse, then it should be easy to see the third point could be
(3,-5) or (-4,2)
#2
give yourself an actual circle with a radius
e.g. r = 10
Draw in different lines and test each case
Make sure your consider the case when your line is a diameter.
in c) did you mean " .... from O is less than 1/2r" ?
#3. your triangle is isosceles with AB = BC, so
3x-8 = x
2x = 8
x = 4
then AC = 4-2 = 2
1. To find the third vertex of a right triangle given two points, you need to check whether the lengths of the sides formed by the two given points satisfy the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
In this case, the two given points are (-4, -5) and (3, 2). Let's label them as A (-4, -5) and B (3, 2). We need to find the possible coordinates for the third vertex, which will be labeled as C (x, y).
First, calculate the length of the sides AB and BC.
AB = sqrt((3 - (-4))^2 + (2 - (-5))^2) = sqrt(49 + 49) = sqrt(98)
BC = sqrt((x - 3)^2 + (y - 2)^2)
Now, we need to check if the length of the side AC satisfies the Pythagorean theorem. AC^2 = BC^2 - AB^2
Substitute the values we know:
(x - 3)^2 + (y - 2)^2 - 98 = 0
At this point, we have an equation with two unknowns (x and y), so we cannot find the exact coordinates of point C. However, we can find multiple potential solutions by plugging in different values for x and solving for y.
2. To answer the second question, let's consider the possible scenarios:
a. The distance between the points of intersection is less than r.
In this scenario, the line l intersects the circle such that the points of intersection fall within the circle. This means that the distance between these two points is definitely less than the radius r.
b. There is at least 1 point on l whose distance from O is less than r.
This statement is true because line l intersects the circle at two distinct points. Hence, at least one of these two points will have a distance from the center O that is less than the radius r.
c. There is no point on l whose distance from O is less than 1/2.
This statement is not necessarily true. Line l can intersect the circle such that one of the points of intersection (or even both) is very close to the center O, which would make the distance less than 1/2 of the radius.
Therefore, option b is the correct answer.
3. In triangle ABC, if the measure of angle A is equal to the measure of angle C, it means that AB is equal to BC in length (since we are given that AB is x and BC is 3x - 8).
So, x = 3x - 8.
Simplifying this equation, we get:
2x = 8
x = 4
Now, since AB = BC, we can substitute the value of x:
AB = BC = 4
However, we need to find the length of AC. Given that AB = x = 4 and AC = x - 2, we can find AC by substituting the value of x:
AC = 4 - 2 = 2
Therefore, the length of AC is 2.