ACT Math - Coordinate Geometry

Math3

What is the measurement of ?

Explanation

Whenever you have an angle that is inscribed to the outside edge of a circle and to an angle that passes through the midpoint of the circle, the inscribed angle will always be one half the measurement of the angle that passes through the midpoint of the circle.

Since the angle that passes through the midpoint of the circle is a straight angle (all straight angles measure degrees), the inscribed angle must measure degrees.

Math3-p1

Since the sum of the internal angles of all triangles add up to degrees, add up the measurements of the angles that you know and subtract the sum from degrees to find your answer:

The coordinates of the endpoints of $\overline{CD}$ , in the standard coordinate plane, are and . What is the -coordinate of the midpoint of $\overline{CD}$ ?

Explanation

To answer this question, we need to find the midpoint of $\overline{CD}$ .

To find how far the midpoint of a line is from each end, we use the following equation:

$midpoint :distance=\frac{x_{2}-x_{1}}{2}, \frac{y_{2}-y_{1}}{2}$

$x_{2}$ and $y_{2}$ are taken from the value of the second point and $x_{1}$ and $y_{1}$ are taken from the value of the first point. Therefore, for this data:

$midpoint :distance=\frac{x_{2}-x_{1}}{2}, \frac{y_{2}-y_{1}}{2}=\frac{7-(-2)}{2}, \frac{1-(-1)}{2}$

We can then solve:

$\frac{7-(-2)}{2}, \frac{1-(-1)}{2}=\frac{9}{2},\frac{2}{2}=4.5,1$

Therefore, our midpoint is units between each endpoint's value and unit between each endpoint's value. To find out the location of the midpoint, we subtract the midpoint distance from the $x_{2},y_{2}$ point. (In this case it's the point .) Therefore:

So the midpoint is located at

The question asked us what the -coordinate of this point was. Therefore, our answer is .

A deer walks in a straight line for 8 hours. At the end of its journey, the deer is 30 miles north and 40 miles east of where it began. What was the average speed of the deer?

miles per hour

Explanation

To find the speed of the deer, you must have the distance traveled and the time.

The distance is found using the Pythagorean Theorem:

$C = {\sqrt{A^2+ B^2}}$

$C = \sqrt{30^2 + 40^2}$

$C = \sqrt{2500} = 50$

The answer must be in miles per hour so the total miles are divided by the hours to get the final answer:

$\frac{miles}{hour} = \frac{50}{8} = 6.25 mph$

Math4

What is the measurement of $\angle B$ ?

Explanation

If you extend the lines of the parellelogram, you will notice that a parellogram is the same as 2 different sets of parellel lines intersecting one another. When that happens, the following angles are congruent to one another:

Math4-p1

Therefore,

Which point satisfies the system and $y = x^{2} - 17$

$\left ( -2,-13\right )$

$\left ( 2,-13\right )$

$\left (7,-32 \right )$

$\left (-7,32 \right )$

None of the other answers

Explanation

In order to solve this problem, we need to find a point that will satisfy both equations. In order to do this, we need to combine the two equations into a single expression. For this, we need to isolate either x or y in one of the equations. Since the equation $y = x^{2} - 17$ already has y isolated, we will use this equation. Next we substitue this equation into the first one. becomes $5x = \left ( x^{2}-17\right )+3$ which simplifies to $x^{2}-5x-14=0$ . Now we can solve for x by factoring: $\left ( x-7\right )\cdot\left ( x+2\right )=0$ Thus, .

Now that we have two possible values for x, we can plug each value into either equation to obtain two values for y. For and the second equation, we get $y = \left ( 7\right )^{2} - 17= 32$ . Therefore our first point is $\left ( 7,32\right )$ . This is not one of the listed answers, so we will use our other value of x. For and the second equation, we get $y = \left ( -2\right )^{2}-17 = -13$ . This gives us the point $\left ( -2,-13\right )$ , which is one of the possible answers.

Find the distance between the points and .

Explanation

The easiest way to find the distance between two points whose coordinates are given in the form and is to use the distance formula.

$d=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}$

Plugging in the coordinates from our given points, our formula looks as follows

$d=\sqrt{(-5-7)^2)+(6-1)^2}$

We then simply simplify step by step

$d=\sqrt{(-12)^2+(5)^2}=\sqrt{144+25}=\sqrt{169}=13$

Therefore, the distance between the two points is 13.

Which of the following is the slope-intercept form of ?

Explanation

To answer this question, we must put the equation into slope-intercept form, meaning we must solve for . Slope-intercept form follows the format where is the slope and is the intercept.

Therefore, we must solve the equation so that is by itself. First we add to both sides so that we can start to get by itself:

$10x+2y-4 +4=0+4\rightarrow 10x+2y=4$

Then, we must subtract from both sides:

$10x+2y -10x=4-10x\rightarrow 2y=-10x+4$

We then must divide each side by

$\frac{2y}{2}=\frac{-10x+4}{2}\rightarrow y=-5x+2$

Therefore, the slope-intercept form of the original equation is .

What is the midpoint between the points $\left ( -1,5\right )$ and $\left ( 7,3\right )$ ?

$\left ( 3,4\right )$

$\left ( 6,8\right )$

$\left ( 4,1\right )$

$\left ( 2,0.5\right )$

$\left ( 3,3\right )$

Explanation

To find the midpoint, find the midpoint (or just average) for the x and y value separately. For the x-value, this means: $\small \small \frac{-1+7}{2}=3$ . For the y-value, this means: $\small \frac{5+3}{2}=4$ . Thus, the midpoint is (3,4).

What is the slope of the line given by the equation ?

$\frac {2}{3}$

$-\frac {2}{3}$

$\frac {3}{2}$

$-\frac {3}{2}$

Explanation

To find the slope, put the line in slope intercept form. In other words put the equation in form where represents the slope and represents the y-intercept.

$y=\frac{2}{3}x+3$

From here we can see our slope equals :

$m=\frac{2}{3}$

Find the distance between $\left ( -1,-2\right )$ and $\left ( 3, -8\right )$

$2\sqrt{13}$

$2\sqrt{10}$

$2\sqrt{29}$

$2\sqrt{26}$

None of the other answers

Explanation

The expression used in solving this question is the distance formula: $d=\sqrt{\left ( x_{2}-x_{1}\right )^{2}+\left ( y_{2}-y_{1}\right )^{2}}$

This formula is simply a variation of the Pythagorian Theorem. A great way to remember this formula is to visualize a right triangle where two of the vertices are the points given in the problem statement. For this question: Distance_between_two_points

Where a = $\left ( x_{2}-x_{1}\right )$ and b = $\left ( y_{2}-y_{1}\right )$ . Now it should be easy to see how the distance formula is simply a variation of the Pythagorean Theorem.

We almost have all of the information we need to solve the problem, but we still need to find the coordinates of the triangle at the right angle. This can be done by simply taking the y-coordinate of the first point and the x-coordinate of the second point, resulting in $\left ( 3,-2\right )$ .