Geometry Regents Review Part 2

Similar Polygons

100

Which of the following is not preserved under a dilation? A.) Angle Measure; B.) Collinearity; C.) Distance; D.) Orientation

C.) Distance

200

In Triangle ABC, D is the midpoint of side AB and E is the midpoint of AC. DE is drawn. If DE = 15, DB = 6, and AD = 10, what is the length of side BC?

Proportion: 10:15 = 16:x; 10x = 240 ; Answer: BC = 24

300

In triangle ABC, the angle measure of angle A is 30 degrees, & the length of segment AB is 12 cm. Determine the length of A'B' & the measure angle A after a dilation D3.

Angle A = 30 degrees A'B' = 36 cm

400

In triangle DEF, point A lies on side DE, and point B lies on side DF. AB is drawn and AB || EF. AD = 6, BD 10, & AE = 3. Determine the length of BF.

Proportion: 6:3 = 10:x; 6x = 30; Answer: BF = 5

500

Which of the following is not a valid method of proving two triangles are similar? A.) SSS ; B.) SAS ; C.) AA ; D.) HL

D.) HL

Circles

100

Chords AB and CD are parallel in a circle. If AB is a diameter, and the measure of arc AC = 40 degrees, then the measure of arc CD is A.) 180; B.) 40; C.) 100; D.) 60

C.) 100 degrees

200

Point P has its vertex outside the circle, and a secant and tangent line are drawn from external point P. to find the measure of angle P you would

Subtract the 2 intercepted arcs and divide by 2

300

Point P lies at the intersection of chord AB and CD. To find the measure of angle APC you would

add the measure of arc AC and arc DB and then divide by 2.

400

A circle is inscribed in triangle ABC such that D,E, and F lie on sides AB, BC, and AC respectively, and points D,E,and F are points on the circle. AD, AE, and BF are all tangent lines to the circle. AD = 6, FC = 8, and BD = 9. What is the length of BC?

BC = 8 + 9 = 17. ( Tangents from the same external point are equal. )

500

Two secants are drawn from point P. The first secant, PAB, PA = 5, and AB = 15. The other secant, PCD, PC = 4. What is the measure of CD?

(Whole secant )x (external piece) = (whole secant x external piece) ; (PA + AB )( PA ) = ( PC + CD )(PC); (5 + 15 )(5) = ( 4 + x )(4) ; 100 = 4x + 16; 4x = 84; Answer: x = 21

Coordinate Geometry

100

Point T(2,-3 ) & Point S(1, -5). What are the coordinates of point M, the midpoint of TS?

Midpoint: [ (x1 + x2) / 2 , (y1+y2)/2 ]; Answer: M(1.5, -4 )

200

Point D(-2,-3) and Point E(4, -6 ). What is the length of line segment DE?

Distance Formula: Sqrt of [( x2-x1)^2 + ( y2 - y1)^2 ]; Sqrt of { (6)^2 + (-3)^2 } ; SQRT of ( 36 + 9 ); SQRT of 45 = 3 radical 5

300

A quadrilateral has ordered pairs A(3,3) ; B(5,4); C(-4, 7); and D( 1, -3). Are AB and CD parallel, perpendicular, or neither?

Compare slopes: Slope of AB: (y2-y1)/(x2-x1) ---> 1/2; Slope of CD: 10/-5 = -2/1; Since -2/1 & 1/2 are negative reciprocals of each other, AB is perpendicualr to CD.

400

Triangle ABC is graphed in the coordinate plane with A(0,4), B( 6, 4) and C(3,2). Classify the triangle by its sides. Justify your answer.

Distance formula: AB = SQRT of 36 = 6; BC = SQRT of 13; AC = SQRT of 13. Answer: Triangle ABC is isosceles.

500

Write the equation of a line perpendicular to 3y - 2x = 6, and passes through the point (-6, 2 ).

Slope of given line is 2/3 when you move the x and then divide. Perpendicular slope = -3/2. Substitute in y=mx+b; 2 = -3/2(-6) + b; b = -7. Answer: Equation is y = -3/2x - 7.

Triangles

100

The lengths of the sides of a triangle could be: A.) 4, 5, 8; B.) 16, 20, 40; C.) 2, 3, 5; D.) 1, 1, 3

A.) 4,5,8. Triangle Inequality Theorem. The sum of any two sides must be greater than the third side. 4 + 5 > 8.

200

Medians AG and BE are drawn in triangle ABC. Point E lies on side AC, and point G lies on side BC. The medians intersect at point W. If BE = 24, what is the length of BW?

W is the centroid, and it divides the medians into a 2:1 ratio. So, let BW = 2x, WE = x. ; 2x + x = 24; x = 8, so BW = 16.

300

Triangle ABC, with sides AB = 6, BC = 4, and AC = 7. What is the perimeter of the triangle formed by connecting the midpoints of all the sides of triangle ABC?

Perimeter = 3 + 2 + 3.5 = 8.5

400

Del draws a triangle with side lengths of 0.8, 1.7, and 1.5 cm. Is the triangle a right triangle? Justify your answer.

Yes, by the pythagorean theorem. (0.8)^2 + (1.5)^2 = (1.7)^2

500

Triangle ABC has angle A measuring 43 degrees, and angle B measures 65 degrees. What is the largest side of triangle ABC?

Angle C measures 72 degrees, and is the largest angle. Answer: AB is the largest side b/c it is opposite the largest angle.

Constructions

100

Show an angle bisector construction or explain the steps.

From the vertex, make an arc that crosses both rays. From each point where the arc has crossed both rays, & without changing the compass width, make an x. Connect the vertex to the x with a straightedge.

200

Show how to construct a perpendicular bisector of a segment or explain the steps.

From each endpoint, ( make the compass width a little over halfway between the endpoints ), make an arc above and below the line such that they intersect. Draw a line from each intersection point with a straightedge.

300

Show how to construct an equilateral triangle, or explain the steps, given a line segment, and a working ray.

Measure the line segment with the compass. Draw an arc to show this on the given line segment. Without changing the width of the compass, draw an arc on the ray from the endpoint & draw an arc above the line segment. Without changing the compass width, draw another arc above the ray from where your 1st arc crossed the ray. Connect all intersection points using a straightedge, to form an equilateral triangle.

400

Construct a line perpendicular to a given line through a point P not on the line. ( Or explain how to do the construction.)

From point P, sweep an arc across the given line. From these intersection points, draw an x arc below the line. Connect point P to the x using a straightedge.

500

Show or explain how to construct a line parallel to a given line through point P not on the line.

Draw a transversal through point P and the given line with a straightedge. Draw an arc at the intersection of the transversal and the given line. Draw a similar arc at P with the same compass width. Measure the 1st arc with compass, and mark it off with an arc. Do the same thing above, where the transversal intersects the arc you drew. Draw a line through point P and the intersection of both arcs with a straightedge.