University:
Morehouse College
Course:
HMTH 105 | College Algebra 1
Academic year:

2024
Views:

402

Pages:

2
Author:

Reynaldo Wong

Question 1 The perimeter of a triangle is the sum of its side lengths, so we first need to find the three side lengths of the triangle. Since the side from (-3, -1) to (2, -1) is a horizontal segment, we can count the number of units between the two points to find its length. Since 2 − ( − 3 ) = 5 2−(−3)=5, then the length of this side is 5. Since the side from (2, -1) to (2, 3) is a vertical segment, we can count the number of units between the two points to find its length. Since 3 − ( − 1 ) = 4 3−(−1)=4, then the length of this side is 4. The side between (-3, -1) to (2, 3) is not a horizontal or vertical segment, so we need to use the distance formula 𝑑 = ( 𝑥 2 − 𝑥 1 ) 2 + ( 𝑦 2 − 𝑦 1 ) 2 d= (x 2 −x 1 ) 2 +(y 2 −y 1 ) 2 to find the length of this side: 𝑑 = ( − 3 − 2 ) 2 + ( − 1 − 3 ) 2 d= (−3−2) 2 +(−1−3) 2 Substitute: = ( − 5 ) 2 + ( − 4 ) 2 = (−5) 2 +(−4) 2 Subtract: = 25 + 16 = 25+16 Evaluate the powers: = 41 = 41 Add. The length of the third side is then 41 ≈ 6.4 41 ≈6.4. The perimeter of the triangle is then about 5 + 4 + 6.4 = 15.4 5+4+6.4=15.4. The exact perimeter is: 5 + 4 + 41 = 9 + 41 5+4+ 41 =9+ 41 Question 2 The base of a triangle is 3 ft more than the height. If the area is 14 ft², find the base and the height. Answer 1 The area of a triangle with base 𝑏 b and height ℎ h is given by: 𝐴 = 1 2 𝑏 ℎ A= 2 1 bh We are given that 𝑏 = ℎ + 3 b=h+3, so we have: 14 = 1 2 ( ℎ + 3 ) ℎ 14= 2 1 (h+3)h Multiply both sides by 2: 28 = ℎ 2 + 3 ℎ 28=h 2 +3h Subtract 28 from both sides: 0 = ℎ 2 + 3 ℎ − 28 0=h 2 +3h−28 Factor the right side: 0 = ( ℎ + 7 ) ( ℎ − 4 ) 0=(h+7)(h−4) By zero product property: ℎ = − 7 , 4 h=−7,4 The height cannot be negative, so we have: ℎ = 4 ft h=4ft. Which follows that: 𝑏 = 4 + 3 = 7 ft b=4+3=7ft. Question 3 [Triangle] Find the value of 𝑥 x and 𝑦 y. Answer 1 When a line divides a triangle such that the line is parallel to a side of the triangle, then it divides the other sides proportionally. The horizontal line is parallel to the base of the triangle, so it divides the left and right sides of the triangle proportionally. Therefore: 𝑥 16 = 45 20 16 x = 20 45 The line with the double arrows is parallel to the right side of the triangle, so it divides the left side and base proportionally. Therefore: 𝑥 16 = 45 20 and 𝑥 16 = 45 𝑦 then 𝑦 20 = 45 𝑦 16 x = 20 45 and 16 x = y 45 then 20 y = y 45 Solving for 𝑦 y gives: 𝑦 2 = 20 ( 45 ) y 2 =20(45) 𝑦 2 = 900 y 2 =900 𝑦 = 900 y= 900 𝑦 = 30 y=30 Substitute 𝑦 = 30 y=30 into either of the original proportions and solve for 𝑥 x: 𝑥 16 = 30 20 16 x = 20 30 𝑥 = 30 20 ⋅ 16 = 3 2 ⋅ 16 x= 20 30 ⋅16= 2 3 ⋅16 𝑥 = 24 x=24