Question

1. which point is always located inside an obtuse triangle? a. the mid - point of one of the sides b. a point outside the triangle c. centroid d. circumcenter 24. what is the standard form of the equation for a circle with center at (2, - 3) and radius 8? a. (x - 2)^2+(y + 3)^2 = 64 b. (x + 2)^2+(y - 3)^2 = 64 c. (x + 2)^2+(y - 3)^2 = 8 d. (x - 2)^2+(y + 3)^2 = 8 25. a circular garden has a radius of 12 meters. if you want to fence a sector with a central angle of 90°, what is the length of the fence along the arc? a. 24 meters b. 12 meters c. 6π meters d. 12π meters

Explanation:

23.

The centroid of a triangle is the point of intersection of the medians and is always inside the triangle, regardless of whether it is obtuse, acute, or right - angled. The mid - point of a side may or may not be inside depending on the triangle's shape. A point outside the triangle is clearly not inside. The circum - center of an obtuse triangle is outside the triangle.

Step1: Recall the standard form of a circle equation

The standard form of a circle equation is $(x - h)^2+(y - k)^2=r^2$, where $(h,k)$ is the center of the circle and $r$ is the radius.

Step2: Identify the values of $h$, $k$, and $r$

Given the center $(2,-3)$ and radius $r = 8$, so $h = 2$, $k=-3$.

Step3: Substitute the values into the equation

Substituting into $(x - h)^2+(y - k)^2=r^2$ gives $(x - 2)^2+(y+3)^2=64$.

Step1: Recall the arc - length formula

The formula for the length of an arc of a circle is $s=r\theta$, where $r$ is the radius of the circle and $\theta$ is the central angle in radians.

Step2: Convert the angle from degrees to radians

Since $90^{\circ}=\frac{\pi}{2}$ radians and $r = 12$ meters.

Step3: Calculate the arc - length

Substitute $r = 12$ and $\theta=\frac{\pi}{2}$ into $s=r\theta$, we get $s=12\times\frac{\pi}{2}=6\pi$ meters.

Answer:

c. Centroid

24.