Answer: The height is: 10.61 units. {or, write as: "(7.5 √2) units"}. _____________________________________________________ Explanation: _____________________________________________________ Area of a triangle: A = 1/2 * base * height = 1/2 * b * h ;
Given: b = 1.5 * h ; A = 75 units² ______________________________ Solve for "h" ("height") ; h = 1.5 b ; _________________________________ So, we solve for "b" ; then we plug that value into the equation:
→ h = 1.5 b ; to get the height, "h" ; ______________________________________________________ → 75 = 1/2 * (b) (1.5 b) ; ______________________________________________________ Multiply the ENTIRE equation by "2" ; to get rid of the fraction and decimals ; ______________________________________________________ → 2 * { 75 = 1/2 * (b) (1.5 b) } ; ______________________________________________________ → 150 = b * 3b ; ______________________________________________________ → 150 = 3 b² ;
Now, divide EACH side of the equation by "3" ; _________________________________________________ → 3 b² / 3 = 150 / 3 ; _________________________________________________ → b² = 50 ; _________________________________________________ Now, take the POSITIVE square root of each side of the equation; to isolate "b" on one side of the equation; and to solve for "b" ; _________________________________________________ → √(b²) = √50 <span>_________________________________________________ </span> → b = √50 _________________________________________________ → √50 = √25 *√2 = 5√2 _________________________________________________ h = (1.5) b = (1.5) *(5) ( _____________________________________________________ Explanation: _____________________________________________________ Area of a triangle: A = 1/2 * base * height = 1/2 * b * h ;
Given: b = 1.5 h ; A = 75 units² ______________________________ Solve for "h" ("height") ; h = 1.5 b ; _________________________________ So, we solve for "b" ; then we plug that value into the equation:
→ h = 1.5 b ; to get the height, "h" ; ______________________________________________________ → 75 = 1/2 * (b) (1.5 b) ; ______________________________________________________ Multiply the ENTIRE equation by "2" ; to get rid of the fraction and decimals ; ______________________________________________________ → 2 * { 75 = 1/2 * (b) (1.5 b) } ; ______________________________________________________ → 150 = b * 3b ; ______________________________________________________ → 150 = 3 b² ;
Now, divide EACH side of the equation by "3" ; _________________________________________________ → 3 b² / 3 = 150 / 3 ; _________________________________________________ → b² = 50 ; _________________________________________________ Now, take the POSITIVE square root of each side of the equation; to isolate "b" on one side of the equation; and to solve for "b" ; _________________________________________________ → √(b²) = √50 _________________________________________________ → b = √50 _________________________________________________ → b = √50 = √25 *√2 = 5√2 _________________________________________________ → h = 1.5 * 5 * √2 ; _________________________________________________ → h = (7.5 √2) units ; or, (7.5) * (√2) = 10.6066017177982129 units ; _______________________________________________________ → round to 10.61 units . _______________________________________________________
The directrix is horizontal, so the axis of symmetry is vertical. We'll have an x^2 term. The vertical distance from y = 4 to y = 8 is 4 units. Cut this in half to get 2, which is the focal distance p = 2.
The point (2,4) is directly below (2,8), and the point is on the directrix. The midpoint between (2,4) and (2,8) is (2,6). This is the vertex.
