Unit 7 Lesson 5: Coordinate PRoofs

Question 1

isosceles

Answer 1

having two congruent sides

Question 2

radius

Answer 2

line from the center to the circumference of a circle

Question 3

scalene

Answer 3

having no sides equal in length

Question 4

Circle P has center P(3,0) , and its radius is 10. Prove that the point Q(11,6) lies on the circumference circle P.

Answer 4

Solution
If point Q
lies on circle P
, PQ¯¯¯¯¯
must be a radius of the circle. In other words, you must prove that PQ¯¯¯¯¯=10
.

PQ¯¯¯¯¯=(11−3)2+(6−0)2−−−−−−−−−−−−−−−−√=82+62−−−−−−√=64+36−−−−−−√=100−−−√=10

Therefore, PQ¯¯¯¯¯=10
, which proves that point Q
lies on circle P
.

Question 5

How can you prove if a point is on a circle when given the length of the radius and coordinates of the center?

Answer 5

Every point that is on a circle is the same distance from the center of the circle. Calculate the distance of a given point from the center of a circle. If the distance is the same as the radius, then the point is on the circle.

Question 6

How can you prove a triangle is isosceles when you are given the coordinates of its vertices?

Answer 6

An isosceles triangle has two legs that are the same length and one length that is different. Using the distance formula, the length of each side length can be calculated from the coordinates. If two of the distances are the same, then the triangle is an isosceles triangle.

Question 7

Prove that the points E(−4,2)
, F(3,−5)
, and G(5,4)
form an isosceles triangle.

Answer 7

Use the distance formula to determine the side lengths and show that △EFG
has two congruent sides.

EF=((−4)−3)2+(2−(−5))2−−−−−−−−−−−−−−−−−−−−−√=(−7)2+72−−−−−−−−−√=49+49−−−−−−√=98−−√≈9.9FG=(3−5)2+((−5)−4)2−−−−−−−−−−−−−−−−−−√=(−2)2+(−9)2−−−−−−−−−−−−√=4+81−−−−−√=85−−√≈9.2GE=(5−(−4))2+(4−2)2−−−−−−−−−−−−−−−−−−√=92+22−−−−−−√=81+4−−−−−√=85−−√≈9.2

Question 8

Properties of Trapezoids

Answer 8

one pair of parallel sides

Question 9

Properties of Kites

Answer 9

• no parallel sides
• diagonals are perpendicular
• one diagonal bisects the opposite angle

Question 10

Properties of Parallelograms

Answer 10

• opposite sides are parallel
• opposite sides are congruent
• opposite angles are congruent
• consecutive angles are supplementary

Question 11

Properties of a Rectangle

Answer 11

• opposite sides are congruent
• opposite sides are parallel
• all four angles are equal
• opposite angles are congruent
• all angles are right angles
• diagonals are congruent

Question 12

Properties of a Rhombus

Answer 12

• all four sides are congruent
• opposite sides are parallel
• opposite sides are congruent
• opposite angles are congruent
• diagonals are perpendicular

Question 13

Properties of a Square

Answer 13

• all four sides are congruent
• opposite sides are congruent
• opposite sides are parallel
• opposite angles are congruent
• all angles are right angles