Math / Geometry

Question20. In the figure, find the coordinates of the mid-point of $A B$ . A. $\left(-\frac{7}{2},-\frac{35}{2}\right)$ B. $\left(-\frac{5}{2},-\frac{25}{4}\right)$ C. $\left(-\frac{5}{2},-\frac{37}{2}\right)$ D. $\left(\frac{7}{2}, \frac{35}{2}\right)$
By Ameina 20 Maple Tutorial Centre

Studdy Solution

STEP 1

What is this asking? We need to find the middle of the line segment connecting the two points where a parabola and a straight line cross! Watch out! Don't mix up the x and y coordinates, and be careful with those negative signs!

STEP 2

1. Find the Intersection Points
2. Calculate the Midpoint

STEP 3

To find where the parabola $y = -x^2$ and the line $y = 5x - 6$ intersect, we set them equal to each other!
This gives us $-x^2 = 5x - 6$ .

STEP 4

Let's rearrange our equation to solve for x.
Add $x^2$ to both sides to get $0 = x^2 + 5x - 6$ .
Now we have a nice quadratic equation!

STEP 5

We're looking for two numbers that multiply to -6 and add up to 5.
Those numbers are 6 and -1!
So, we can factor our equation as $0 = (x + 6)(x - 1)$ .

STEP 6

This gives us two possible solutions for x: $x = -6$ and $x = 1$ .
These are the x-coordinates of our intersection points, A and B!

STEP 7

Now, plug these x values back into either equation (let's use the line equation because it's simpler) to find the corresponding y values. For $x = -6$ , $y = 5 \cdot (-6) - 6 = -30 - 6 = -36$ . For $x = 1$ , $y = 5 \cdot 1 - 6 = 5 - 6 = -1$ . So, our intersection points are $(-6, -36)$ and $(1, -1)$ .

STEP 8

The midpoint of a line segment with endpoints $(x_1, y_1)$ and $(x_2, y_2)$ is given by $\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$ .
It's like finding the average of the x values and the average of the y values!

STEP 9

Let's plug in our coordinates $(-6, -36)$ and $(1, -1)$ into the midpoint formula: $\left( \frac{-6 + 1}{2}, \frac{-36 + (-1)}{2} \right)$

STEP 10

$\left( \frac{-5}{2}, \frac{-37}{2} \right)$ So, the midpoint is $\left( -\frac{5}{2}, -\frac{37}{2} \right)$ !

STEP 11

The midpoint of AB is $\left( -\frac{5}{2}, -\frac{37}{2} \right)$ , which is answer choice C!

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Question20. In the figure, find the coordinates of the mid-point of $A B$ . A. $\left(-\frac{7}{2},-\frac{35}{2}\right)$ B. $\left(-\frac{5}{2},-\frac{25}{4}\right)$ C. $\left(-\frac{5}{2},-\frac{37}{2}\right)$ D. $\left(\frac{7}{2}, \frac{35}{2}\right)$
By Ameina 20 Maple Tutorial Centre

Studdy Solution

STEP 1

What is this asking? We need to find the middle of the line segment connecting the two points where a parabola and a straight line cross! Watch out! Don't mix up the x and y coordinates, and be careful with those negative signs!

STEP 2

1. Find the Intersection Points
2. Calculate the Midpoint

STEP 3

To find where the parabola $y = -x^2$ and the line $y = 5x - 6$ intersect, we set them equal to each other!
This gives us $-x^2 = 5x - 6$ .

STEP 4

Let's rearrange our equation to solve for x.
Add $x^2$ to both sides to get $0 = x^2 + 5x - 6$ .
Now we have a nice quadratic equation!

STEP 5

We're looking for two numbers that multiply to -6 and add up to 5.
Those numbers are 6 and -1!
So, we can factor our equation as $0 = (x + 6)(x - 1)$ .

STEP 6

This gives us two possible solutions for x: $x = -6$ and $x = 1$ .
These are the x-coordinates of our intersection points, A and B!

STEP 7

STEP 8

The midpoint of a line segment with endpoints $(x_1, y_1)$ and $(x_2, y_2)$ is given by $\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$ .
It's like finding the average of the x values and the average of the y values!

STEP 9

Let's plug in our coordinates $(-6, -36)$ and $(1, -1)$ into the midpoint formula: $\left( \frac{-6 + 1}{2}, \frac{-36 + (-1)}{2} \right)$

STEP 10

$\left( \frac{-5}{2}, \frac{-37}{2} \right)$ So, the midpoint is $\left( -\frac{5}{2}, -\frac{37}{2} \right)$ !

STEP 11

The midpoint of AB is $\left( -\frac{5}{2}, -\frac{37}{2} \right)$ , which is answer choice C!

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Studdy Solution

STEP 1

What is this asking? We need to find the *middle* of the line segment connecting the two points where a parabola and a straight line cross! Watch out! Don't mix up the *x* and *y* coordinates, and be careful with those negative signs!

STEP 2

1. Find the Intersection Points2. Calculate the Midpoint

STEP 3

To find where the parabola y=−x2y = -x^2y=−x2 and the line y=5x−6y = 5x - 6y=5x−6 intersect, we set them equal to each other!This gives us −x2=5x−6-x^2 = 5x - 6−x2=5x−6.

STEP 4

Let's rearrange our equation to solve for *x*.Add x2x^2x2 to both sides to get 0=x2+5x−60 = x^2 + 5x - 60=x2+5x−6.Now we have a nice quadratic equation!

STEP 5

We're looking for two numbers that multiply to **-6** and add up to **5**.Those numbers are **6** and **-1**!So, we can factor our equation as 0=(x+6)(x−1)0 = (x + 6)(x - 1)0=(x+6)(x−1).

STEP 6

This gives us two possible solutions for *x*: x=−6x = -6x=−6 and x=1x = 1x=1.These are the *x*-coordinates of our intersection points, **A** and **B**!

STEP 7

STEP 8

STEP 9

Let's plug in our coordinates (−6,−36)(-6, -36)(−6,−36) and (1,−1)(1, -1)(1,−1) into the midpoint formula: (−6+12,−36+(−1)2)\left( \frac{-6 + 1}{2}, \frac{-36 + (-1)}{2} \right)(2−6+1​,2−36+(−1)​)

STEP 10

(−52,−372)\left( \frac{-5}{2}, \frac{-37}{2} \right)(2−5​,2−37​) So, the midpoint is (−52,−372)\left( -\frac{5}{2}, -\frac{37}{2} \right)(−25​,−237​)!

STEP 11

The midpoint of AB is (−52,−372)\left( -\frac{5}{2}, -\frac{37}{2} \right)(−25​,−237​), which is answer choice C!

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What is this asking? We need to find the middle of the line segment connecting the two points where a parabola and a straight line cross! Watch out! Don't mix up the x and y coordinates, and be careful with those negative signs!

1. Find the Intersection Points
2. Calculate the Midpoint

To find where the parabola $y = -x^2$ and the line $y = 5x - 6$ intersect, we set them equal to each other!
This gives us $-x^2 = 5x - 6$ .

Let's rearrange our equation to solve for x.
Add $x^2$ to both sides to get $0 = x^2 + 5x - 6$ .
Now we have a nice quadratic equation!

We're looking for two numbers that multiply to -6 and add up to 5.
Those numbers are 6 and -1!
So, we can factor our equation as $0 = (x + 6)(x - 1)$ .

This gives us two possible solutions for x: $x = -6$ and $x = 1$ .
These are the x-coordinates of our intersection points, A and B!

Let's plug in our coordinates $(-6, -36)$ and $(1, -1)$ into the midpoint formula: $\left( \frac{-6 + 1}{2}, \frac{-36 + (-1)}{2} \right)$

$\left( \frac{-5}{2}, \frac{-37}{2} \right)$ So, the midpoint is $\left( -\frac{5}{2}, -\frac{37}{2} \right)$ !

The midpoint of AB is $\left( -\frac{5}{2}, -\frac{37}{2} \right)$ , which is answer choice C!