BM 2023

Consider two circles, circle X and Circle Y shown on the Graph.

Which of the following statements describes how circles X can be shown to be similar to circle Y, by mapping circle X onto circle Y?

• Translate circle X by multiplying the x-value and y-value of the center of circle X by 2, and then dilate circle X by a scale factor of 2.

• Translate circle X by adding 2 units to the xvalue and y-value of the center of circle X, and then dilate circle X by a scale factor of 2

• Translate circle X by adding 3 units to the xvalue and subtracting 3 units from the y-value of the center of circle X, and then dilate circle X by a scale factor of 2.

• Translate circle X by adding 3 units to the xvalue and subtracting 3 units from the y-value of the center of circle X, and then dilate circle Y by a scale factor of 2.

Which strategy can be used to prove that the diagonals of a parallelogram bisect each other.

• Find the slope of each diagonal and prove that the slopes are negative reciprocals of each other.

• Find the slopes of two adjacent sides of the parallelogram and prove that they are negative reciprocals of each other.

• Calculate the length of each diagonal and show that half of one diagonal is as long as half of the other diagonal.

Answer explanation

Determining that the midpoints of the two diagonals are the same proves that they bisect each other.

Three highways connect the centers of three towns and form a triangle. A cell phone company wants to place a new cell tower so that it is the same distance from the centers of the three towns. How can the company find where to place the tower?

• The company must find the point of intersection of the medians for the triangle.

• The company must find the point of intersection of the altitudes for the triangle.

• The company must find the point of intersection of the angle bisectors for the triangle

• The company must find the point of intersection of the perpendicular bisectors for the triangle.

Answer explanation

The company must find the point that is equidistant from the vertices of the triangle which is the circumcenter of the triangle. This can be found by finding the point of intersection of the perpendicular bisectors of the triangle.

Answer explanation

The radius of the circle is 3, so r^2= 9 . Substituting the x- and y-values of the given point into the equation form of a circle

Jackson has a small triangular piece of cardboard. He wants to balance the triangle on the end of his pencil. Which of these triangles shows the intersection that contains the balance point Jackson can use to balance the triangle on his pencil?

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Answer explanation

Jackson must find the point that is the center of gravity of the triangle. This point will be the intersection of the medians of the triangle.

Line a is perpendicular to line b, and neither line is vertical. Which statement(s) below are TRUE?

• I: The slopes of line a and line b are the same.

• II: One line has a positive slope, the other line has a negative slope.

• III: Line a and line b will never intersect.

• IV: The slope of one of the lines is undefined.

Two circles centered at the origin are graphed on the coordinate plane. The smaller circle has a radius of 2 units and the larger circle has a radius of 8 units.

Select all of the transformations that will prove that the two circles are similar.

• Translate the larger circle 6 units to the left.

• Translate the smaller circle 6 units up

• Dilate the smaller circle by a scale factor of 4about its origin.

• Dilate the smaller circle by a scale factor of 6about its origin.

A triangle has vertices at ( − 2, − 3), (4, − 3), and (3, 5). What is the area of the triangle?

• 18 units squared

• 24 units squared

• 36 units squared

• 48 units squared

Answer explanation

This is the result of first recognizing that the height of the triangle is equivalent to the vertical distance formed by the line segment extending from (–2, –3) to (3, 5). The length of this segment is calculated by subtracting –3 (the y-coordinate of (–2, –3)) from the ycoordinate of the vertex (3, 5) to yield 5 – (– 3) = 8 units. The base of the triangle is calculated as the difference of the xcoordinates between (–2, –3) and (4, –3), which is 4 – (–2) = 6 units. Therefore, the area is given by (1/2)(6 units)(8 units) = 24 units2.

A city planner designed a circular bike path for the local park. The coordinate grid shows the bike path she designed.

What is the equation of the circle that represents the bike path?

Answer explanation

The circle has a center of (-2,1) and a radius of 4.