Distance Formula Vicente Rosalia Alcantara NIE/CTP- Math

Direction: Read carefully the following five multiple choice questions. Choose the correct answer for each number by clicking the triangle beside each option.

1. Points A and B are on the same line. If A(x 1, y 1 ) and B(x 2, y 2 ), find the length of line segment AB ? a. AB = √ (x 2 + x 1 ) 2 - ( y 2 + y 1 ) 2 b. AB = √ (x 2 - x 1 ) 2 + ( y 2 - y 1 ) 2 c. AB = √(x 1 + y 1 ) 2 - ( x 2 + y 2 ) 2 d. AB = √(x 1 - y 1 ) 2 + ( x 2 - y 2 ) 2

CORRECT In Cartesian Coordinate Plane, a line can be presented by defining the coordinates of its two endpoints, i.e. for line AB,pt. A(x 1, y 1 ) and B(x 2, y 2 ). By knowing the two endpoints of a line, we can find the length of the line segment or the distance between two points. B (x 2, y 2 ) CA(x 1,y 1 ) y x 0 By Pythagorean Theorem AB 2 = AC 2 + BC 2 = ( x 2 -x 1 ) 2 + ( y 2,y 1 ) 2 = √( x 2 -x 1 ) 2 + ( y 2,y 1 ) 2 Note: When we calculate the length of a line AB, we can interchange the coordinate values of A and B, and we still get the same value for length AB.

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2. If A and B has a coordinates (1,1) and (4,5) respectively, find the distance between the two given points. a. √ 61 units b. 25 units c. 5 units d. 1 unit

CORRECT Solution: If A (1,1) and B (4,5), then x 1 =1 ; y 1 =1 and x 2 =4 ; y 2 =5 By Pythagorean Theorem AB 2 = AC 2 + BC 2 = √( x 2 - x 1 ) 2 + ( y 2,y 1 ) 2 = √ (4 – 1) 2 + (5 - 1) 2 = √ (3) 2 +(4) 2 = √ = √ 25 = 5 Note: When we calculate the length of a line AB, we can interchange the coordinate values of A and B, and we still get the same value for length AB.

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3. If points A(-2,8), B(0,y) and C( 2,0) lie on the same straight line, find the value of y if AB = BC. a. 4 b. √8 c. 16 d. √20

CORRECT Solution: a. First, find the length of AB and BC in terms of y; Length of AB 2 = [0-(-2)] 2 + (y-8) 2 Length of BC 2 = (2-0) 2 +(0-y) 2 AB 2 = (y-8) BC 2 = y AB = √(y – 8) BC = √ y b. Then solve for y; √(y – 8) = √ y y 2 – 16y + 64 = y 2 16y = 64 ; y = 4 c. Therefore; AB = √(y – 8) BC =√ y AB = √(4 -8) BC = √ AB = √20 BC =√ 20

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4. Points X,Y and Z are the vertices of Isosceles Triangle XYZ. If the coordinates of X, Y and Z are (2,3),(-6,3) and(-2,0) respectively, which of its sides are congruent to each other? a. XZ and XY b. XY and YZ c. YZ and XZ d. none of the above

CORRECT Solution: a.To determine which of the three sides are congruent, find the length of each sides by using distance formula. XY=√[(-6)-2] 2 +(3-3) 2 YZ=√[(-2)-(-6)] 2 +(0-3) 2 XZ=√[(-2)-2] 2 +(0-3) 2 XY=√(-8) 2 +(0) 2 YZ=√(4) 2 +(-3) 2 XZ=√(-4) 2 +(-3) 2 XY=√64 YZ=√16+9 XZ=√16+9 XY= 8 YZ= 5 XZ= 5 b. Therefore, the sides of Isosceles Triangle XYZ that are congruent to each other are: YZ and XZ

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5. Find the perimeter of the triangle XYZ, given that its vertices X,Y and Z are ( 1,4), (2,0) and (-1,4) respectively. a. 2 b. 5 c. 7 √17 d. 17 √7

CORRECT Solution: a.To get the perimeter of Triangle XYZ, find the length of the three sides of triangle XYZ by distance formula. XY =√(1-2) 2 +(4-0) 2 YZ = √[2-(-1)] 2 +(0-4) 2 XZ = √[1-(-1)] 2 +(4-4) 2 XY =√(-1) 2 +(4) 2 YZ = √(3) 2 +(-4) 2 XZ = √(2) 2 +(0) 2 XY = √ YZ = √ XZ = √ 4 XY = √ 17 YZ = 5 XZ = 2 b. Therefore; the Perimeter of Triangle XYZ = XY + YZ + XZ = √ = 7 √ 17

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