501 Geometry Questions

Set 93

454. You could draw the figure, or you could find the slope between

each line. The slope of ABis (–5(2––(– 61 )))^44 1. The slope of BC

is (–1(6––(– 6 )^5 )) 04 . The slope of CAis (–5(6––(– 25 ))), or ^04 . BCis vertical

because its slope is undefined; CAis horizontal because its slope

equals zero. Horizontal and vertical lines meet perpendicularly;

therefore ΔABC is a right triangle.

455. Again, you could draw figure ABCD in a coordinate plane and

visually confirm that it is a parallelogram, or you could find the

slope and distance between each point. The slope of ABis

(–( 83 – –(–^56 ))), or ^22 . The distance between •A and •B is (2)^2 + (2)^2 ,

or 2 2 . The slope of BCis ((–^56 – –^54 )), or –1^0  0. The distance between •B

and •C is the difference of the xcoordinates, or 10. The slope of CD

is ((^54 – – 23 )), or ^22 . The distance between •C and •D is  2 ^2 + 2^2 , or 2 2 .

The slope of line DAis ((–^38 – –^32 )), or ^010 . The distance between •D

and •A is the difference of the x-coordinates, or 10. From the

calculations above you know that opposite ABand CDhave the

same slope and length, which means they are parallel and con-

gruent. Also opposite lines BCand DAhave the same zero slope

and lengths; again, they are parallel and congruent; therefore

figure ABCD is a parallelogram because opposite sides AB/CDand

BC/DAare parallel and congruent.

456. You must prove that only one pair of opposite sides in figure

ABCD is parallel and noncongruent. Slope AB is – 0 ^4 ; its length is

the difference of ycoordinates, or 4. Slope BCis –^0  4 ; its length is

the difference of xcoordinates, or 4. Slope of CDis ^60 ; its length is

the difference of ycoordinates, or 6. Finally, slope of DAis (– 21 ); its

length is  4 ^2 + (– 2 ^2 ), or 2 5 . Opposite sides ABand CDhave the

same slope but measure different lengths; therefore they are

parallel and noncongruent. Figure ABCD is a trapezoid.

501 Geometry Questions