the three points g(4,0) ,h(h,6) and i(7,1) are such that gh is twice as long as gi .calculate the two possible values of h.
using distance formula
d of GI= square root of 10-3.1622
twice this is the square root of 40
could be- 6 or 2
To calculate the two possible values of "h," we need to determine the lengths of both GH and GI and find the value of "h" that satisfies the condition where GH is twice as long as GI.
Let's start by finding the lengths of GH and GI:
Length of GH:
To find the length of GH, we use the distance formula:
Distance formula: √((x₂ - x₁)² + (y₂ - y₁)²)
GH = √((h - 4)² + (6 - 0)²)
Length of GI:
Similarly, we use the distance formula to find the length of GI:
GI = √((7 - 4)² + (1 - 0)²)
According to the given condition, GH is twice as long as GI:
GH = 2 * GI
Now, we can substitute the formulas we derived into the equation and solve for "h":
√((h - 4)² + 36) = 2 * √13
Squaring both sides of the equation to eliminate the square root:
((h - 4)² + 36) = 4 * 13
(h - 4)² + 36 = 52
(h - 4)² = 52 - 36
(h - 4)² = 16
h - 4 = ±√16
h - 4 = ±4
Now, we solve for "h" in each case:
Case 1: h - 4 = 4
h = 4 + 4
h = 8
Case 2: h - 4 = -4
h = -4 + 4
h = 0
Therefore, the two possible values of "h" are 8 and 0.
To calculate the two possible values of h, we can use the distance formula to find the lengths of both gh and gi. Since gh is twice as long as gi, we can set up an equation and solve for h.
Let's start by finding the lengths of gh and gi using the distance formula:
The distance between two points (x₁, y₁) and (x₂, y₂) is given by the formula:
d = √((x₂ - x₁)² + (y₂ - y₁)²)
For gh:
x₁ = 4, y₁ = 0 (g(4,0))
x₂ = h, y₂ = 6 (h(h,6))
The length of gh (gh_length) is given by:
gh_length = √((h - 4)² + (6 - 0)²)
For gi:
x₁ = 4, y₁ = 0 (g(4,0))
x₂ = 7, y₂ = 1 (i(7,1))
The length of gi (gi_length) is given by:
gi_length = √((7 - 4)² + (1 - 0)²)
Now, let's set up the equation using the information given:
gh_length = 2 * gi_length
√((h - 4)² + (6 - 0)²) = 2 * √((7 - 4)² + (1 - 0)²)
Simplifying the equation, we have:
√((h - 4)² + 36) = 2 * √(9 + 1)
Squaring both sides of the equation:
((h - 4)² + 36) = 4 * 10
Expanding and simplifying:
(h - 4)² + 36 = 40
(h - 4)² = 40 - 36
(h - 4)² = 4
Take the square root of both sides:
h - 4 = ±2
h = 4 ± 2
Therefore, the two possible values of h are 6 and 2.
gi^2 = 3^2+1^2 = 10^2
gh^2 = 4 * 10^2
4*10^2 = (h-4)^2 + 36
400 = h^2 - 8h + 36
h^2 - 8 h - 364 = 0
h = [ 8 +/- sqrt(64+1456) ]/2
h = 4 +/- 19.5
h = 23.5 or -15.5
check my arithmetic !